A276537 Alternating binomial sums of the cubes of the central binomial coefficients.

1, 7, 201, 7375, 312265, 14365887, 697859169, 35226348087, 1829569294665, 97138289500735, 5248514415816721, 287657066913117447, 15953440327189548001, 893653778439275931175, 50488236061157830951545, 2873526763346873838886815

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..558

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} (-1)^(n-k)*binomial(n,k)*binomial(2*k,k)^3.

Recurrence: (n^3+12*n^2+48*n+64)*a(n+4)-(60*n^3+630*n^2+2204*n+2569)*a(n+3)-(186*n^3+1674*n^2+5037*n+5067)*a(n+2)-94*(2*n^3+15*n^2+37*n+30)*a(n+1)-63*(n^3+6*n^2+11*n+6)*a(n)=0.

G.f.: (4/Pi^2)*K(1/2-1/2*sqrt((1-63*t)/(1+t)))^2)/(1+t), where K(x) is the complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).

a(n) ~ 3^(2*n+3) * 7^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 16 2016

a(n) = (-1)^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(-1)^(n-k), {k, 0, n}], {n, 0, 100}]

Prog

(Maxima) makelist(sum(binomial(n, k)*binomial(2*k, k)^3*(-1)^(n-k), k, 0, n), n, 0, 12);
(Magma) [&+[(-1)^(n-k)*Binomial(n, k)*Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 03 2016

Crossrefs

Cf. A276536.

Similar sums of m-powers of the central binomial coefficients: A002426 (m=1), A278934 (m=2), this sequence (m=3).

Sequence in context: A220934 A221288 A355087 * A233507 A316862 A226414

Adjacent sequences: A276534 A276535 A276536 * A276538 A276539 A276540

Keyword

nonn

Author

Emanuele Munarini, Nov 16 2016

Status

approved