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Search: a217487 -id:a217487
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A217484
Partial sums of the numbers in sequence A080253.
+10
6
1, 4, 21, 168, 1865, 26348, 450205, 9011152, 206624529, 5338349652, 153408637349, 4853054571896, 167576795780953, 6271355892192316, 252836327218276653, 10924378168890333600, 503589353964709474337, 24669610145575233317540
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
FORMULA
a(n) = sum(c(k),k=0..n), where c(n) = A080253(n).
E.g.f.: exp (x)/(2-exp(2*x)) + x*exp (x)/2 + (1/4)*exp(x)*log(1/(2-exp(2*x))). - corrected by Vaclav Kotesovec, Jan 02 2013
a(n) ~ n! * 2^(n-1/2)/(log(2))^(n+1). - Vaclav Kotesovec, Jan 02 2013
MATHEMATICA
t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[c[k], {k, 0, n}], {n, 0, 100}]
PROG
(Maxima) t(n):=sum(stirling2(n, k)*k!, k, 0, n);
c(n):=sum(binomial(n, k)*2^k*t(k), k, 0, n);
makelist(sum(c(k), k, 0, n), n, 0, 10);
CROSSREFS
Cf. A080253, A000670, A217483, A217485, A217486, A217487, A217488.
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Oct 04 2012
STATUS
approved
A217486
Binomial convolution of the numbers in sequence A080253.
+10
6
1, 6, 52, 600, 8656, 149856, 3026752, 69866880, 1814338816, 52350752256, 1661575754752, 57531530434560, 2158011794968576, 87173881613869056, 3772959800981143552, 174183372619165040640, 8543978588021450407936, 443748799382401230176256
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = sum(binomial(n,k)*c(k)*c(n.k),k=0..n), where c(n) = A080253(n).
a(n) = 2^n*t(n+1), where t(n) = ordered Bell numbers (A000670).
E.g.f. exp(2*x)/(2-exp(2*x))^2.
G.f.: 1/G(0) where G(k) = 1 - x*3*(2*k+2) + x^2*(k+1)*(k+2)*(1-3^2)/G(k+1) ; (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013.
a(n) ~ n!*n*2^(n-1)/(log(2))^(n+2). - Vaclav Kotesovec, Aug 11 2013
MATHEMATICA
t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[Binomial[n, k]c[k]c[n-k], {k, 0, n}], {n, 0, 100}]; Table[2^n t[n+1], {n, 0, 100}]
With[{nn=20}, CoefficientList[Series[Exp[2x]/(2-Exp[2x])^2, {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Oct 09 2017 *)
PROG
(Maxima) t(n):=sum(stirling2(n, k)*k!, k, 0, n);
c(n):=sum(binomial(n, k)*2^k*t(k), k, 0, n);
makelist(sum(binomial(n, k)*c(k)*c(n-k), k, 0, n), n, 0, 10);
makelist(2^n*t(n+1), n, 0, 40);
(Sage)
def A217486(n):
return 2^n*add(add((-1)^(j-i)*binomial(j, i)*i^(n+1) for i in range(n+2)) for j in range(n+2))
[A217486(n) for n in range(18)] # Peter Luschny, Jul 22 2014
CROSSREFS
Cf. A080253, A000670, A217483, A217484, A217485, A217487, A217488.
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Oct 04 2012
STATUS
approved
A217483
Alternating sums of the numbers in sequence A080253.
+10
5
1, 2, 15, 132, 1565, 22918, 400939, 8160008, 189453369, 4942271754, 143128015943, 4556517918604, 158167223290453, 5945611873120910, 240619359452963427, 10430922482219093520, 482234053313600047217, 23683786738296923795986
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..17.
FORMULA
a(n) = sum((-1)^(n-k)*c(k),k=0..n), where c(n) = A080253(n).
E.g.f.: exp(x)/(2-exp(2*x)) - (1/2)*exp(-x)*log(1/(2-exp(2*x))). - corrected by Vaclav Kotesovec, Nov 27 2017
a(n) ~ n! * 2^(n - 1/2) / (log(2))^(n+1). - Vaclav Kotesovec, Nov 27 2017
MATHEMATICA
t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[(-1)^(n-k)c[k], {k, 0, n}], {n, 0, 100}]
nmax = 20; CoefficientList[Series[E^x/(2 - E^(2*x)) + Log[2 - E^(2*x)] / (2*E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 27 2017 *)
PROG
(Maxima) t(n):=sum(stirling2(n, k)*k!, k, 0, n);
c(n):=sum(binomial(n, k)*2^k*t(k), k, 0, n);
makelist(sum((-1)^(n-k)*c(k), k, 0, n), n, 0, 10);
CROSSREFS
Cf. A080253, A000670, A217484, A217485, A217486, A217487, A217488.
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Oct 04 2012
STATUS
approved
A217485
Convolution of the numbers in sequence A080253.
+10
5
1, 6, 43, 396, 4565, 64146, 1073919, 20996376, 471081385, 11947911966, 338204687315, 10570101018276, 361458024882045, 13421571912745386, 537661560385125031, 23108777539028187696, 1060571767117824260945, 51760585513634983767606
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..17.
FORMULA
a(n) = sum(c(k)*c(n.k),k=0..n), where c(n) = A080253(n).
a(n) ~ n! * 2^(n + 1/2) / (log(2))^(n+1). - Vaclav Kotesovec, Nov 27 2017
MATHEMATICA
t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[c[k]c[n-k], {k, 0, n}], {n, 0, 100}]
PROG
(Maxima) t(n):=sum(stirling2(n, k)*k!, k, 0, n);
c(n):=sum(binomial(n, k)*2^k*t(k), k, 0, n);
makelist(sum(c(k)*c(n-k), k, 0, n), n, 0, 40);
CROSSREFS
Cf. A080253, A000670, A217483, A217484, A217486, A217487, A217488.
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Oct 04 2012
STATUS
approved
A217488
Alternating sums of the squares of the numbers in sequence A080253
+10
5
1, 8, 281, 21328, 2858481, 596558808, 179058197641, 73110755339168, 38977936014004961, 26295624802015360168, 21898514473870334203641, 22064773395630274673891568, 26456951179676525013504937681, 37229662306608638451691410580088
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..13.
FORMULA
a(n) = sum((-1)^(n-k)*c(k)^2,k=0..n), where c(n) = A080253(n).
a(n) ~ (n!)^2 * 2^(2*n-1) / (log(2))^(2*n + 2). - Vaclav Kotesovec, Nov 27 2017
MATHEMATICA
t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[(-1)^(n-k)c[k]^2, {k, 0, n}], {n, 0, 100}]
PROG
(Maxima) t(n):=sum(stirling2(n, k)*k!, k, 0, n);
c(n):=sum(binomial(n, k)*2^k*t(k), k, 0, n);
makelist(sum((-1)^(n-k)*c(k)^2, k, 0, n), n, 0, 40);
CROSSREFS
Cf. A080253, A000670, A217483, A217484, A217485, A217486, A217487.
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Oct 04 2012
STATUS
approved
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Last modified October 25 21:03 EDT 2024. Contains 377104 sequences.
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