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
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Nov 16 2016
STATUS
approved