A275822 Alternating sums of the cubes of the central binomial coefficients.

1, 7, 209, 7791, 335209, 15667799, 773221225, 39651016343, 2092095886657, 112840936041343, 6193764391911873, 344853399798469695, 19429178297906958721, 1105629520934309041279, 63455683531507986958721, 3668895994183490904049279

Offset: 0

Author

Emanuele Munarini, Nov 15 2016

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..555
• The Wolfram Functions Site, Complete Elliptic Integrals

Crossrefs

Cf. A079727.

Programs

Maple

L:= [seq((-1)^k*binomial(2*k,k)^3,k=0..20)]:
B:= ListTools:-PartialSums(L):
seq((-1)^(k+1)*B[k],k=1..nops(B)); # Robert Israel, Nov 21 2016

Mathematica

Table[Sum[Binomial[2 k, k]^3 (-1)^(n - k), {k, 0, n}], {n, 0, 20}]
Table[Sum[(-1)^(n - k) (k + 1)^3 CatalanNumber[k]^3, {k, 0, n}], {n, 0, 20}] (* Jan Mangaldan, Jul 07 2020 *)

Maxima

makelist(sum(binomial(2*k,k)^3*(-1)^(n-k),k,0,n),n,0,12);

PARI

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k,k)^3); \\ Michel Marcus, Jul 07 2020

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k,k)^3.

Recurrence: (n+2)^3*a(n+2)-(3*n+4)*(21*n^2+66*n+52)*a(n+1)-8*(2n+3)^3*a(n)=0.

G.f.: (4/Pi^2)*K((1-sqrt(1-64*t))/2)^2/(1+t), where K(x) is complete elliptic integral of the first kind (defined as in The Wolfram Functions Site).

a(n) ~ 2^(6*n+6) / (65*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Nov 16 2016