A248586 a(n) = Sum_{i=0..n} C(n,i)*C(2i,i)^2.

1, 5, 45, 521, 6733, 92385, 1316865, 19274925, 287694285, 4359037985, 66837293545, 1034774126325, 16149186405025, 253737607849445, 4009771017244485, 63681603585696321, 1015763347140335565, 16264070907887454465

Offset: 0

Author

R. J. Mathar, Oct 09 2014

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..815

Crossrefs

Cf. A002894 (inverse binomial transform), A002893.

Programs

Mathematica

Table[Sum[Binomial[n, k] Binomial[2k, k]^2, {k, 0, n}],{n,0,100}] (* Emanuele Munarini, Oct 28 2016 *)

Maxima

makelist(sum(binomial(n,k)*binomial(2*k,k)^2,k,0,n),n,0,12); /* Emanuele Munarini, Oct 28 2016 */

PARI

a(n) = sum(i=0, n, binomial(n,i)*binomial(2*i,i)^2); \\ Michel Marcus, Oct 09 2014

Formula

a(n) = Sum_{i=0..n} A007318(n,i)*A002894(i).

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