OFFSET
0,8
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).
FORMULA
a(n) = binomial(n/2,3)^2, n even;
a(n) = binomial((n-1)/2,3)*binomial((n+1)/2,3), n odd.
From Colin Barker, Jan 21 2020: (Start)
G.f.: x^6*(1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^7*(1 + x)^5).
a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>11.
(End)
E.g.f.: (cosh(x)-sinh(x))*(45+36*x+18*x^2+6*x^3+3*x^4+(-45+54*x-36*x^2+18*x^3-9*x^4+6*x^5+2*x^6)*(cosh(2*x)+sinh(2*x)))/4608. - Stefano Spezia, Jan 27 2020
EXAMPLE
a(7) = 4 and the 4 subsets are {1,2,3,4,5,6}, {1,2,3,4,6,7}, {1,2,4,5,6,7}, {2,3,4,5,6,7}.
MAPLE
a:= n-> ((b, q)-> b(q, 3)*b(n-q, 3))(binomial, iquo(n, 2)):
seq(a(n), n=0..50); # Alois P. Heinz, Jan 30 2020
MATHEMATICA
a[n_] := If[OddQ[n], Binomial[(n - 1)/2, 3]*Binomial[(n + 1)/2, 3], Binomial[n/2, 3]^2]; Array[a, 45, 0] (* Amiram Eldar, Jan 21 2020 *)
LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {0, 0, 0, 0, 0, 0, 1, 4, 16, 40, 100, 200}, 50] (* Harvey P. Dale, Dec 17 2022 *)
PROG
(PARI) concat([0, 0, 0, 0, 0, 0], Vec(x^6*(1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^7*(1 + x)^5) + O(x^40))) \\ Colin Barker, Jan 21 2020
(Magma) [IsEven(n) select Binomial((n div 2), 3)^2 else Binomial((n-1) div 2, 3)*Binomial((n+1) div 2, 3): n in [0..45]]; // Marius A. Burtea, Jan 21 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jan 20 2020
STATUS
approved