A213409 G.f.: exp( Sum_{n>=1} binomial(3*n^2,n^2) * x^n/n ).

1, 3, 252, 1563022, 563716946982, 10517711119760250261, 9692061982207456039533424586, 430311348543725825536505706371595438684, 906895928239445077568583988067142630846220290783969, 89857639488565787203362892584824012528872539028234934088960440084

Offset

0,2

Comments

Compare to the g.f. G(x) = 1 + x*G(x)^3 of A001764: G(x)^3 = exp( Sum_{n>=1} binomial(3*n,n) * x^n/n ).

Links

G. C. Greubel, Table of n, a(n) for n = 0..34

Formula

a(n) = (1/n) * Sum_{k=1..n} binomial(3*k^2,k^2) * a(n-k) for n>0 with a(0)=1.

a(n) ~ sqrt(3) * (27/4)^(n^2) / (2*sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014

Example

G.f.: A(x) = 1 + 3*x + 252*x^2 + 1563022*x^3 + 563716946982*x^4 +...
where
log(A(x)) = 3*x + 495*x^2/2 + 4686825*x^3/3 + 2254848913647*x^4/4 + 52588547141148893628*x^5/5 +...+ C(3*n^2,n^2)*x^n/n +...

Mathematica

nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*k^2, k^2]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(3*m^2, m^2)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(3*k^2, k^2)*a(n-k)))}

Crossrefs

Cf. A201556, A213410, A001764, A245245 (log).

Sequence in context: A025418 A075901 A028918 * A227028 A279653 A320023

Adjacent sequences: A213406 A213407 A213408 * A213410 A213411 A213412

Keyword

nonn

Author

Paul D. Hanna, Jun 10 2012

Status

approved