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

0,2

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.

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

Mathematica

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

Prog

(Maxima) makelist(sum(binomial(n, k)*binomial(2*k, k)^3, k, 0, n), n, 0, 12);
(Magma) [&+[Binomial(n, k)*Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 30 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).

Sequence in context: A230178 A165389 A153223 * A288683 A302914 A157569

Adjacent sequences: A276533 A276534 A276535 * A276537 A276538 A276539

Keyword

nonn

Author

Emanuele Munarini, Nov 16 2016

Status

approved