A276536 Binomial sums of the cubes of the central binomial coefficients.

1, 9, 233, 8673, 376329, 17800209, 890215361, 46294813497, 2478150328777, 135642353562321, 7556884938829233, 427106589765940137, 24429206859151618209, 1411391470651692285609, 82245902444586364980057, 4828398428680134702936273

Offset: 0

Author

Emanuele Munarini, Nov 16 2016

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..554
• Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind
• The Wolfram Functions Site, Complete Elliptic Integrals, 2016.

Crossrefs

Cf. Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^m: A026375 (m=1), A248586 (m=2), this sequence (m=3).

Programs

Magma

[&+[Binomial(n, k)*Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 30 2016

Mathematica

Table[Sum[Binomial[n, k]Binomial[2k, k]^3, {k, 0, n}], {n, 0, 100}]

Maxima

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

Formula

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

Recurrence: (n^3 + 12n^2 + 48n + 64) * a(n+4) - (68n^3 + 714n^2 + 2500n + 2919) * a(n+3) + (198n^3 + 1782n^2 + 5363n + 5397) * a(n+2) - 98 * (2n^3 + 15n^2 + 37n + 30) * a(n+1) + 65 * (n^3 + 6n^2 + 11n + 6) * a(n) = 0.

G.f.: (4/Pi^2) * K(1/2 - 1/2 * sqrt((1-65*t)/(1-t)))^2 / (1-t), where K(x) is complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).

a(n) ~ 65^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 16 2016

a(n) = 4F3(1/2,1/2,1/2,-n; 1,1,1; -64). - Ilya Gutkovskiy, Nov 25 2016