Displaying 1-10 of 22 results found.
Eigentriangle generated from A109128, row sums = expansion of {2(exp(x)-1)}
+20
1
1, 1, 1, 1, 3, 2, 1, 5, 10, 6, 1, 7, 22, 42, 22, 1, 9, 38, 114, 198, 94, 1, 11, 58, 234, 638, 1034, 454, 1, 13, 82, 414, 1518, 3854, 5902, 2430, 1, 15, 110, 666, 3058, 10434, 24970, 36450, 14214, 1, 17, 142, 1002, 5522, 23594, 75818, 172530, 241638, 89918
COMMENTS
Row sums = A001861: (1, 2, 6, 22, 94, 454, 2430,...) = expansion of {2(exp(x)-1)} Right border = A001861 shifted: (1, 1, 2, 6, 22, 94,...). Sum of n-th row terms = rightmost term of next row.
FORMULA
A109128 = (2*binomial(n,k) - 1): (1; 1,1; 1,3,1; 1,5,5,1;...).
A001861(k-1) = A001861 shifted one place, = (1, 1, 2, 6, 22, 94, 454,...).
EXAMPLE
First few rows of the triangle =
1;
1, 1;
1, 3, 2;
1, 5, 10, 6;
1, 7, 22, 42, 22;
1, 9, 38, 114, 198, 94;
1, 11, 58, 234, 638, 1034, 454;
1, 13, 82, 414, 1518, 3854, 5902, 2430;
1, 15, 110, 666, 3058, 10434, 24970, 36450, 14214;
...
Example: row 3 = (1, 5, 10, 6) = termwise products of (1, 5, 5, 1) and (1, 1, 2, 6), where (1, 5, 5, 1) = row 3 of triangle A109128 and (1, 1, 2, 6) = the first 4 terms of A001861 shifted.
A triangle of polynomial coefficients: p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= (less than or equal to) Floor[n/2], (-1)^i* A109128[n, i], -(-1)^(n - i)* A109128[n, i]]], {i, 0, n}]/(1 - x).
+20
0
1, 1, 1, 1, -4, 1, 1, -6, -6, 1, 1, -8, 11, -8, 1, 1, -10, 19, 19, -10, 1, 1, -12, 29, -40, 29, -12, 1, 1, -14, 41, -70, -70, 41, -14, 1, 1, -16, 55, -112, 139, -112, 55, -16, 1, 1, -18, 71, -168, 251, 251, -168, 71, -18, 1, 1, -20, 89, -240, 419, -504, 419, -240, 89, -20, 1
COMMENTS
Row sums are:
{1, 2, -2, -10, -3, 20, -4, -84, -5, 274, -6,...}.
FORMULA
p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= (less than or equal to) Floor[n/2], (-1)^i* A109128[n, i], -(-1)^(n - i)* A109128[n, i]]], {i, 0, n}]/(1 - x); t(n,m)=coefficients(p(x,n),x)
EXAMPLE
{1},
{1, 1},
{1, -4, 1},
{1, -6, -6, 1},
{1, -8, 11, -8, 1},
{1, -10, 19, 19, -10, 1},
{1, -12, 29, -40, 29, -12, 1},
{1, -14, 41, -70, -70, 41, -14, 1},
{1, -16, 55, -112, 139, -112, 55, -16, 1},
{1, -18, 71, -168, 251, 251, -168, 71, -18, 1},
{1, -20, 89, -240, 419, -504, 419, -240, 89, -20, 1}
MATHEMATICA
Clear[A, p, n, i];
A[n_, 0] := 1;
A[n_, n_] := 1;
A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + 1;
p[x_, n_] = Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], (-1)^i*A[n, i], -(-1)^(n - i)*A[n, i]]], {i, 0, n}]/(1 - x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]
1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069, 2161, 2255, 2351, 2449, 2549, 2651
COMMENTS
Values of Fibonacci polynomial n^2 - n - 1 for n = 2, 3, 4, 5, ... - Artur Jasinski, Nov 19 2006 a(n-1) gives the number of n X k rectangles on an n X n chessboard (for k = 1, 2, 3, ..., n). - Aaron Dunigan AtLee, Feb 13 2009 sqrt(a(0) + sqrt(a(1) + sqrt(a(2) + sqrt(a(3) + ...)))) = sqrt(1 + sqrt(5 + sqrt(11 + sqrt(19 + ...)))) = 2. - Miklos Kristof, Dec 24 2009 When n + 1 is prime, a(n) gives the number of irreducible representations of any nonabelian group of order (n+1)^3. - Andrew Rupinski, Mar 17 2010 The product of any 4 consecutive integers plus 1 is a square (see A062938); the terms of this sequence are the square roots. - Harvey P. Dale, Oct 19 2011 Or numbers not expressed in the form m + floor(sqrt(m)) with integer m. - Vladimir Shevelev, Apr 09 2012 Another expression involving phi = (1 + sqrt(5))/2 is a(n) = (n + phi)(n + 1 - phi). Therefore the numbers in this sequence, even if they are prime in Z, are not prime in Z[phi]. - Alonso del Arte, Aug 03 2013 a(n-1) = n*(n+1) - 1, n>=0, with a(-1) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 5 for b = 2*n+1. In general D = b^2 - 4ac > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013 An example of a quadratic sequence for which the continued square root map (see A257574) produces the number 2. There are infinitely many sequences with this property - another example is A028387. See Popular Computing link. - N. J. A. Sloane, May 03 2015 The numbers represented as 131 in base n: 131_4 = 29, 131_5 = 41, ... . If 'digits' larger than the base are allowed then 131_2 = 11 and 131_1 = 5 also. - Ron Knott, Nov 14 2017 Let m be a(n) or a prime factor of a(n). Then, except for 1 and 5, there are, if m is a prime, exactly two squares y^2 such that the difference y^2 - m contains exactly one pair of factors {x,z} such that the following applies: x*z = y^2 - m, x + y = z with
x < y, where {x,y,z} are relatively prime numbers. {x,y,z} are the initial values of a sequence of the Fibonacci type. Thus each a(n) > 5, if it is a prime, and each prime factor p > 5 of an a(n) can be assigned to exactly two sequences of the Fibonacci type. a(0) = 1 belongs to the original Fibonacci sequence and a(1) = 5 to the Lucas sequence.
But also the reverse assignment applies. From any sequence (f(i)) of the Fibonacci type we get from its 3 initial values by f(i)^2 - f(i-1)*f(i+1) with f(i-1) < f(i) a term a(n) or a prime factor p of a(n). This relation is also valid for any i. In this case we get the absolute value |a(n)| or |p|. (End)
a(n-1) = 2*T(n) - 1, for n>=1, with T = A000217, is a proper subsequence of A089270, and the terms are 0,-1,+1 (mod 5). - Wolfdieter Lang, Jul 05 2019 a(n+1) is the number of wedged n-dimensional spheres in the homotopy of the neighborhood complex of Kneser graph KG_{2,n}. Here, KG_{2,n} is a graph whose vertex set is the collection of subsets of cardinality 2 of set {1,2,...,n+3,n+4} and two vertices are adjacent if and only if they are disjoint. - Anurag Singh, Mar 22 2021 (x, y, z) = ( A001105(n+1), -a(n-1), -a(n)) are solutions of the Diophantine equation x^3 + 4*y^3 + 4*z^3 = 8. - XU Pingya, Apr 25 2022 The least significant digit of terms of this sequence cycles through 1, 5, 1, 9, 9. - Torlach Rush, Jun 05 2024
LINKS
Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy. Zdzislaw Skupień and Andrzej Żak, Pair-sums packing and rainbow cliques, in Topics In Graph Theory, A tribute to A. A. and T. E. Zykovs on the occasion of A. A. Zykov's 90th birthday, ed. R. Tyshkevich, Univ. Illinois, 2013, pages 131-144, (in English and Russian).
FORMULA
a(0) = 1, a(1) = 5, a(n) = (n+1)*a(n-1) - (n+2)*a(n-2) for n > 1. - Gerald McGarvey, Sep 24 2004 Binomial transform of [1, 4, 2, 0, 0, 0, ...] = (1, 5, 11, 19, ...). - Gary W. Adamson, Sep 20 2007 G.f.: (1+2*x-x^2)/(1-x)^3. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - R. J. Mathar, Jul 11 2009 a(n) = (n + 2 + 1/phi) * (n + 2 - phi); where phi = 1.618033989... Example: a(3) = 19 = (5 + .6180339...) * (3.381966...). Cf. next to leftmost column in A162997 array. - Gary W. Adamson, Jul 23 2009 For k < n, a(n) = (k+1)*a(n-k) - k*a(n-k-1) + k*(k+1); e.g., a(5) = 41 = 4*11 - 3*5 + 3*4. - Charlie Marion, Jan 13 2011 a(n) = lower right term in M^2, M = the 2 X 2 matrix [1, n; 1, (n+1)]. - Gary W. Adamson, Jun 29 2011 G.f.: (x^2-2*x-1)/(x-1)^3 = G(0) where G(k) = 1 + x*(k+1)*(k+4)/(1 - 1/(1 + (k+1)*(k+4)/G(k+1)); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012 Product_{n>=0} (1 + 1/a(n)) = -Pi*sec(sqrt(5)*Pi/2).
Product_{n>=1} (1 - 1/a(n)) = -Pi*sec(sqrt(5)*Pi/2)/6. (End)
EXAMPLE
Illustration of initial terms:
o o
o o o o o o
o o o o o o o o o o o o
o o o o o o o o o o o o o o o o o o o o
o o o o o o o o o o o o o o o o o o o o o o o o o
n=0 n=1 n=2 n=3 n=4
(End)
Examples:
a(0) = 1: 1^1-0*1 = 1, 0+1 = 1 (Fibonacci A000045). a(1) = 5: 3^2-1*4 = 5, 1+3 = 4 (Lucas A000032). a(2) = 11: 4^2-1*5 = 11, 1+4 = 5 ( A000285); 5^2-2*7 = 11, 2+5 = 7 ( A001060). a(3) = 19: 5^2-1*6 = 19, 1+5 = 6 ( A022095); 7^2-3*10 = 19, 3+7 = 10 ( A022120). a(4) = 29: 6^2-1*7 = 29, 1+6 = 7 ( A022096); 9^2-4*13 = 29, 4+9 = 13 ( A022130). a(11)/5 = 31: 7^2-2*9 = 31, 2+7 = 9 ( A022113); 8^2-3*11 = 31, 3+8 = 11 ( A022121). a(24)/11 = 59: 9^2-2*11 = 59, 2+9 = 11 ( A022114); 12^2-5*17 = 59, 5+12 = 17 ( A022137). (End)
MATHEMATICA
Table[ FrobeniusNumber[{n, n + 1}], {n, 2, 30}] (* Zak Seidov, Jan 14 2015 *)
PROG
(Haskell)
(Python) def a(n): return (n**2+3*n+1) # Torlach Rush, May 07 2024
EXTENSIONS
Minor edits by N. J. A. Sloane, Jul 04 2010, following suggestions from the Sequence Fans Mailing List
1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556, 2097131, 4194282, 8388585, 16777192, 33554407, 67108838, 134217701, 268435428, 536870883, 1073741794, 2147483617
COMMENTS
Number of permutations of degree n with at most one fall; called "Grassmannian permutations" by Lascoux and Sch�tzenberger. - Axel Kohnert (Axel.Kohnert(AT)uni-bayreuth.de)
Number of different permutations of a deck of n cards that can be produced by a single shuffle. [DeSario]
Number of Dyck paths of semilength n having at most one long ascent (i.e., ascent of length at least two). Example: a(4)=12 because among the 14 Dyck paths of semilength 4, the only paths that have more than one long ascent are UUDDUUDD and UUDUUDDD (each with two long ascents). Here U = (1, 1) and D = (1, -1). Also number of ordered trees with n edges having at most one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004 Number of {12,1*2*,21*}-avoiding signed permutations in the hyperoctahedral group.
Number of 1342-avoiding circular permutations on [n+1].
2^n - n is the number of ways to partition {1, 2, ..., n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths at least 1. - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005
if b(0) = x and b(n) = b(n-1) + b(n-1)^2*x^(n-2) for n > 0, then b(n) is a polynomial of degree a(n). - Michael Somos, Nov 04 2006 The chromatic invariant of the Mobius ladder graph M_n for n >= 2. - Jonathan Vos Post, Aug 29 2008 Dimension sequence of the dual alternative operad (i.e., associative and satisfying the identity xyz + yxz + zxy + xzy + yzx + zyx = 0) over the field of characteristic 3. - Pasha Zusmanovich, Jun 09 2009 An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 26 and 176, lead to this sequence (without the first leading 1). For the central square these vectors lead to the companion sequence A168604. - Johannes W. Meijer, Aug 15 2010 a(n+1) is also the number of order-preserving and order-decreasing partial isometries (of an n-chain). - Abdullahi Umar, Jan 13 2011 A040001(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
A130103(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
The number of labeled graphs with n vertices whose vertex set can be partitioned into a clique and a set of isolated points. - Alex J. Best, Nov 20 2012 See coefficients of the linear terms of the polynomials of the table on p. 10 of the Getzler link. - Tom Copeland, Mar 24 2016 Consider n points lying on a circle, then for n>=2 a(n-1) is the maximum number of ways to connect two points with non-intersecting chords. - Anton Zakharov, Dec 31 2016 Also the number of cliques in the (n-1)-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017 Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k). [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i), e(j), e(k) pairwise distinct. [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(j) >= e(k) and e(i) != e(k) pairwise distinct. [Martinez and Savage, 2.7]
(End)
Number of F-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are F-equivalent iff the positions of pattern F are identical in these paths. - Sergey Kirgizov, Apr 08 2018 Also the number of connected partitions of an n-cycle. For example, the a(1) = 1 through a(4) = 12 connected partitions are:
{{1}} {{12}} {{123}} {{1234}}
{{1}{2}} {{1}{23}} {{1}{234}}
{{12}{3}} {{12}{34}}
{{13}{2}} {{123}{4}}
{{1}{2}{3}} {{124}{3}}
{{134}{2}}
{{14}{23}}
{{1}{2}{34}}
{{1}{23}{4}}
{{12}{3}{4}}
{{14}{2}{3}}
{{1}{2}{3}{4}}
(End)
Number of subsets of n-set without the single-element subsets. - Yuchun Ji, Jul 16 2019 For every prime p, there are infinitely many terms of this sequence that are divisible by p (see IMO Compendium link and Doob reference). Corresponding indices n are: for p = 2, even numbers A299174; for p = 3, A047257; for p = 5, A349767. - Bernard Schott, Dec 10 2021
REFERENCES
Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.
LINKS
Marty Getz, Dixon Jones and Ken Dutch, Partitioning by Arithmetic Progressions: Problem 11005, American Math. Monthly, Vol. 112, 2005, p. 89. (The published solution is incomplete. Letting d be the common difference of the arithmetic progressions, the solver's expression q_1(n,d)=2^(n-d) must be summed over all d=1,...,n and duplicate partitions must be removed.) The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983. Eric Weisstein's World of Mathematics, Clique
FORMULA
Binomial transform of 1, 0, 1, 1, 1, .... The sequence starting 1, 2, 5, ... has a(n) = 1 + n + 2*Sum_{k=2..n} binomial(n, k) = 2^(n+1) - n - 1. This is the binomial transform of 1, 1, 2, 2, 2, 2, .... a(n) = 1 + Sum_{k=2..n} C(n, k). - Paul Barry, Jun 06 2003 G.f.: 1 / (1 - x / (1 - x / ( 1 - x / (1 + x / (1 - 2*x))))). - Michael Somos, May 12 2012 a(n) = [x^n](B(x)^n-B(x)^(n-1)), n>0, a(0)=1, where B(x) = (1+2*x+sqrt(1+4*x^2))/2. - Vladimir Kruchinin, Mar 07 2014 a(n)^2 - 4*a(n-1)^2 = (n-2)*(a(n)+2*a(n-1)). - Yuchun Ji, Jul 13 2018
EXAMPLE
G.f. = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 58*x^6 + 121*x^7 + ...
MAPLE
A000325 := proc(n) option remember; if n <=1 then n+1 else 2* A000325(n-1)+n-1; fi; end;
g:=1/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-n, n=0..31); # Zerinvary Lajos, Jan 09 2009
MATHEMATICA
LinearRecurrence[{4, -5, 2}, {1, 2, 5}, {0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
PROG
(Haskell)
a000325 n = 2 ^ n - n
a000325_list = zipWith (-) a000079_list [0..]
(Python)
AUTHOR
Rosario Salamone (Rosario.Salamone(AT)risc.uni-linz.ac.at)
Expansion of e.g.f. exp(2*(exp(x) - 1)).
(Formerly M1662 N0653)
+10
77
1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182, 4412798, 33827974, 273646526, 2326980998, 20732504062, 192982729350, 1871953992254, 18880288847750, 197601208474238, 2142184050841734, 24016181943732414, 278028611833689478, 3319156078802044158, 40811417293301014150
COMMENTS
Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.
The number of ways of putting n labeled balls into a set of bags and then putting the bags into 2 labeled boxes. An example is given below. - Peter Bala, Mar 23 2013 The f-vectors of n-dimensional hypercube are given by A038207 = exp[M*B(.,2)] = exp[M* A001861(.)] where M = A238385-I and (B(.,x))^n = B(n,x) are the Bell polynomials (cf. A008277). - Tom Copeland, Apr 17 2014
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = Sum_{k=0..n} 2^k*Stirling2(n, k). - Emeric Deutsch, Oct 20 2001 G.f. satisfies 2*(x/(1-x))*A(x/(1-x)) = A(x) - 1; twice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003 PE = exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1]. - Gottfried Helms, Apr 08 2007 G.f.: 1/(1-2x-2x^2/(1-3x-4x^2/(1-4x-6x^2/(1-5x-8x^2/(1-6x-10x^2/(1-... (continued fraction). - Paul Barry, Apr 29 2009 O.g.f.: Sum_{n>=0} 2^n*x^n / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Feb 15 2012 a(n) ~ exp(-2-n+n/LambertW(n/2))*n^n/LambertW(n/2)^(n+1/2). - Vaclav Kotesovec, Jan 06 2013 G.f.: (G(0) - 1)/(x-1)/2 where G(k) = 1 - 2/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013 G.f.: 1/Q(0) where Q(k) = 1 + x*k - x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013 G.f.: ((1+x)/Q(0)-1)/(2*x), where Q(k) = 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013 G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-2*x-x*k)*(1-3*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013 a(n) = Sum_{k=0..n} A033306(n,k) = Sum_{k=0..n} binomial(n,k)*Bell(k)*Bell(n-k), where Bell = A000110 (see Motzkin, p. 170). - Danny Rorabaugh, Oct 18 2015
EXAMPLE
a(2) = 6: The six ways of putting 2 balls into bags (denoted by { }) and then into 2 labeled boxes (denoted by [ ]) are
01: [{1,2}] [ ];
02: [ ] [{1,2}];
03: [{1}] [{2}];
04: [{2}] [{1}];
05: [{1} {2}] [ ];
06: [ ] [{1} {2}].
MAPLE
# second Maple program:
b:= proc(n, m) option remember;
`if`(n=0, 2^m, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
MATHEMATICA
Table[Sum[StirlingS2[n, k]*2^k, {k, 0, n}], {n, 0, 21}] (* Geoffrey Critzer, Oct 06 2009 *) mx = 16; p = 1; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
PROG
(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(2*(exp(x+x*O(x^n))-1)), n))
(PARI) {a(n)=polcoeff(sum(m=0, n, 2^m*x^m/prod(k=1, m, 1-k*x +x*O(x^n))), n)} /* Paul D. Hanna, Feb 15 2012 */ (PARI) {a(n) = sum(k=0, n, 2^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019 (Magma) [&+[2^k*StirlingSecond(n, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 18 2019
CROSSREFS
Cf. A000110, A000587, A002871, A027710, A056857, A068199, A068200, A068201, A078937, A078938, A078944, A078945, A109128, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A213170, A221159, A221176.
Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.
+10
14
0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 9, 9, 4, 0, 0, 5, 14, 19, 14, 5, 0, 0, 6, 20, 34, 34, 20, 6, 0, 0, 7, 27, 55, 69, 55, 27, 7, 0, 0, 8, 35, 83, 125, 125, 83, 35, 8, 0, 0, 9, 44, 119, 209, 251, 209, 119, 44, 9, 0, 0, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 0
COMMENTS
Indexed as a square array A(n,k): If X is an (n+k)-set and Y a fixed k-subset of X then A(n,k) is equal to the number of n-subsets of X intersecting Y. - Peter Luschny, Apr 20 2012
FORMULA
G.f.: x^2*y/((1 - x)*(1 - x*y)*(1 - x*(1 + y))). - Ralf Stephan, Jan 24 2005 T(n, k) = T(n-1, k-1) + T(n-1, k) + 1, 0 < k < n with T(n, 0) = T(n, n) = 0. - Reinhard Zumkeller, Jul 18 2015 If seen as a square array read by antidiagonals the generating function of row n is: G(n) = (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1)). - Peter Luschny, Feb 13 2019 T(n, n-k) = T(n, k).
Sum_{k=0..floor(n/2)} T(n-k, k) = A129696(n-2). Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n-1), where b(n) is the repeating pattern {0, 0, -1, -2, -1, 1, 1, -1, -2, -1, 0, 0}_{n=0..11}, with b(n) = b(n-12). (End)
EXAMPLE
Triangle begins:
0;
0, 0;
0, 1, 0;
0, 2, 2, 0;
0, 3, 5, 3, 0;
0, 4, 9, 9, 4, 0;
0, 5, 14, 19, 14, 5, 0;
0, 6, 20, 34, 34, 20, 6, 0;
...
Seen as a square array read by antidiagonals:
[0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000004 [1] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... A001477 [2] 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, ... A000096 [3] 0, 3, 9, 19, 34, 55, 83, 119, 164, 219, 285, 363, ... A062748 [4] 0, 4, 14, 34, 69, 125, 209, 329, 494, 714, 1000, 1364, ... A063258 [5] 0, 5, 20, 55, 125, 251, 461, 791, 1286, 2001, 3002, 4367, ... A062988 [6] 0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, ... A124089
MAPLE
with(combstruct): for n from 0 to 11 do seq(-1+count(Combination(n), size=m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008 # The rows of the square array:
Arow := proc(n, len) local gf, ser;
gf := (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1));
ser := series(gf, x, len+2): seq((-1)^(n+1)*coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do lprint([n], Arow(n, 12)) od; # Peter Luschny, Feb 13 2019
MATHEMATICA
Table[Binomial[n, k] -1, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)
PROG
(Haskell)
a014473 n k = a014473_tabl !! n !! k
a014473_row n = a014473_tabl !! n
a014473_tabl = map (map (subtract 1)) a007318_tabl
(Magma)
[Binomial(n, k)-1: k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024 (SageMath)
flatten([[binomial(n, k)-1 for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024
Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.
+10
13
1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 21, 13, 1, 1, 17, 37, 37, 17, 1, 1, 21, 57, 77, 57, 21, 1, 1, 25, 81, 137, 137, 81, 25, 1, 1, 29, 109, 221, 277, 221, 109, 29, 1, 1, 33, 141, 333, 501, 501, 333, 141, 33, 1, 1, 37, 177, 477, 837, 1005, 837, 477, 177, 37, 1
COMMENTS
Row sums = A131062: (1, 2, 7, 20, 49, 110, 235, ...); the binomial transform of (1, 1, 4, 4, 4, ...).
FORMULA
G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 21 2007
EXAMPLE
First few rows of the triangle are
1;
1, 1;
1, 5, 1;
1, 9, 9, 1;
1, 13, 21, 13, 1;
1, 17, 37, 37, 17, 1;
1, 21, 57, 77, 57, 21, 1;
...
MAPLE
T := proc (n, k) if k <= n then 4*binomial(n, k)-3 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007
MATHEMATICA
Table[4*Binomial[n, k] -3, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
PROG
(Magma) [4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020 (Sage) [[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020
Triangle read by rows: T(n,k) = 6*binomial(n,k) - 5 for 0 <= k <= n.
+10
13
1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 31, 19, 1, 1, 25, 55, 55, 25, 1, 1, 31, 85, 115, 85, 31, 1, 1, 37, 121, 205, 205, 121, 37, 1, 1, 43, 163, 331, 415, 331, 163, 43, 1, 1, 49, 211, 499, 751, 751, 499, 211, 49, 1, 1, 55, 265, 715, 1255, 1507, 1255, 715, 265, 55, 1
COMMENTS
The matrix inverse starts:
1;
-1, 1;
6, -7, 1;
-66, 78, -13, 1;
1086, -1284, 216, -19, 1;
-23826, 28170, -4740, 420, -25, 1;
653406, -772536, 129990, -11520, 690, -31, 1; - R. J. Mathar, Mar 12 2013
FORMULA
G.f.: (1-z-t*z+6*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007
EXAMPLE
First few rows of the triangle are:
1;
1, 1;
1, 7, 1;
1, 13, 13, 1;
1, 19, 31, 19, 1;
1, 25, 55, 55, 25, 1;
...
MAPLE
T := proc (n, k) if k <= n then 6*binomial(n, k)-5 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007
MATHEMATICA
Table[6*Binomial[n, k]-5, {n, 0, 15}, {k, 0, n}]//Flatten (* Harvey P. Dale, May 15 2016 *)
PROG
(Magma) [6*Binomial(n, k) -5: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020 (Sage) [[6*binomial(n, k) -5 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020
1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 16, 10, 1, 1, 13, 28, 28, 13, 1, 1, 16, 43, 58, 43, 16, 1, 1, 19, 61, 103, 103, 61, 19, 1, 1, 22, 82, 166, 208, 166, 82, 22, 1, 1, 25, 106, 250, 376, 376, 250, 106, 25, 1, 1, 28, 133, 358, 628, 754, 628, 358, 133, 28, 1
COMMENTS
Row sums = A097813: (1, 2, 6, 16, 38, 84, 178, ...).
EXAMPLE
First few rows of the triangle:
1;
1, 1;
1, 4, 1;
1, 7, 7, 1;
1, 10, 16, 10, 1;
1, 13, 28, 28, 13, 1;
1, 16, 43, 58, 43, 16, 1;
...
MATHEMATICA
T[n_, k_] = 3*Binomial[n, k] -2; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Aug 20 2008 *)
PROG
(Magma) [3*Binomial(n, k) -2: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020 (Sage) [[3*binomial(n, k) -2 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020
Triangle read by rows: T(n,k) = 5*binomial(n,k) - 4 for 0 <= k <= n.
+10
12
1, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 16, 26, 16, 1, 1, 21, 46, 46, 21, 1, 1, 26, 71, 96, 71, 26, 1, 1, 31, 101, 171, 171, 101, 31, 1, 1, 36, 136, 276, 346, 276, 136, 36, 1, 1, 41, 176, 416, 626, 626, 416, 176, 41, 1, 1, 46, 221, 596, 1046, 1256, 1046, 596, 221, 46, 1
COMMENTS
Row sums = A131064: (1, 2, 8, 24, 60, 136, 292, ...), the binomial transform of (1, 1, 5, 5, 5, ...).
FORMULA
G.f.: (1-z-t*z+5*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007
EXAMPLE
First few rows of the triangle:
1;
1, 1;
1, 6, 1;
1, 11, 11, 1;
1, 16, 26, 16, 1;
1, 21, 46, 46, 21, 1;
1, 26, 71, 96, 71, 26, 1;
...
MAPLE
T := proc (n, k) if k <= n then 5*binomial(n, k)-4 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007
MATHEMATICA
Table[5*Binomial[n, k] -4, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
PROG
(GAP) Print(Flat(List([0..10], n->List([0..n], k->5*Binomial(n, k)-4)))); # Muniru A Asiru, Feb 21 2019 (Magma) [5*Binomial(n, k) -4: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020 (Sage) [[5*binomial(n, k) -4 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020
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