OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..815
FORMULA
Conjecture: n^2*a(n) +(-19*n^2+19*n-5)*a(n-1) +35*(n-1)^2*a(n-2) -17*(n-1)*(n-2)*a(n-3)=0.
G.f.: LegendreP(-1/2, (1+15x)/(1-17x)) /[sqrt(1-17x)*sqrt(1-x)]. - Corrected by Robert Israel, Oct 28 2016
From Emanuele Munarini, Oct 28 2016: (Start)
a(n) = hypergeometric(1/2,1/2,-n;1,1;-16).
G.f.: A(t) = (2/Pi)*(ellipticK(16*t/(1-t))/(1-t)).
Diff. eq. satisfied by the g.f.: t*(1-t)*(1-18*t+17*t^2)*A''(t)+(1-t)*(1-37*t+68*t^2)*A'(t)-(34*t^2-35*t+5)*A(t)=0.
Remark: the conjectured recurrence for the coefficients a(n) comes from this diff. eq. for A(t).
(End)
a(n) ~ 17^(n+1)/(16*Pi*n). - Vaclav Kotesovec, Oct 30 2016
MATHEMATICA
Table[Sum[Binomial[n, k] Binomial[2k, k]^2, {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Oct 28 2016 *)
PROG
(PARI) a(n) = sum(i=0, n, binomial(n, i)*binomial(2*i, i)^2); \\ Michel Marcus, Oct 09 2014
(Maxima) makelist(sum(binomial(n, k)*binomial(2*k, k)^2, k, 0, n), n, 0, 12); /* Emanuele Munarini, Oct 28 2016 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Oct 09 2014
STATUS
approved