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

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..555

The Wolfram Functions Site, Complete Elliptic Integrals

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

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 *)

Prog

(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

Crossrefs

Cf. A079727.

Sequence in context: A178022 A157775 A300619 * A065819 A241649 A256288

Adjacent sequences: A275819 A275820 A275821 * A275823 A275824 A275825

Keyword

nonn

Author

Emanuele Munarini, Nov 15 2016

Status

approved