A246571 G.f.: Sum_{n>=0} x^n / (1-x)^(4*n+3) * [Sum_{k=0..2*n+1} C(2*n+1,k)^2 * x^k]^2.

1, 6, 39, 340, 3041, 28718, 279987, 2788464, 28256709, 290124182, 3010689527, 31516942060, 332347297141, 3526399820374, 37616896717155, 403127408462816, 4337723615579781, 46842172878701486, 507454305359968827, 5513119883595629556, 60050379276555861857, 655611405802102543086

Offset

0,2

Comments

A bisection of A246563.

Self-convolution of A246573.

Links

Table of n, a(n) for n=0..21.

Vaclav Kotesovec, Recurrence (of order 8)

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-k-j+1,k)^2 * C(k,j)^2.

Example

G.f.: A(x) = 1 + 6*x + 39*x^2 + 340*x^3 + 3041*x^4 + 28718*x^5 + 279987*x^6 +...
where
A(x) = 1/(1-x)^3 * (1 + x)^2 + x/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3)^2
+ x^2/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
+ x^3/(1-x)^15 * (1 + 7^2*x + 21^2*x^2 + 35^2*x^3 + 35^2*x^4 + 21^2*x^5 + 7^2*x^6 + x^7)^2 +...
The square-root of the g.f. is an integer series:
A(x)^(1/2) = 1 + 3*x + 15*x^2 + 125*x^3 + 1033*x^4 + 9385*x^5 + 88531*x^6 + 858739*x^7 + 8517503*x^8 + 85867417*x^9 +...+ A246573(n)*x^n +...

Mathematica

Table[Sum[Sum[Binomial[2*n-k-j+1, k]^2 * Binomial[k, j]^2, {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 02 2014 *)

Prog

(PARI) /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(4*m+3) * sum(k=0, 2*m+1, binomial(2*m+1, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
(PARI) /* From a formula for a(n): */
{a(n)=sum(k=0, n, sum(j=0, min(k, 2*n-2*k+1), binomial(2*n-k-j+1, k)^2 * binomial(k, j)^2 ))}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A246563, A246570, A246572, A246573.

Sequence in context: A113347 A265953 A356335 * A031972 A356439 A308861

Adjacent sequences: A246568 A246569 A246570 * A246572 A246573 A246574

Keyword

nonn

Author

Paul D. Hanna, Aug 30 2014

Status

approved