A136638 a(n) = Sum_{k=0..[n/2]} C(n-k, k) * C(3^(n-2*k)*2^k, n-k).

1, 3, 38, 2955, 1666194, 6775599252, 204212962736426, 47025953519744215608, 84798028785462127288681736, 1219731316443261012339196962784452, 141916030637329352970764084182705691263552

Offset: 0

Author

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

Keywords

• nonn

Comments

Equals antidiagonal sums of triangle A136635.

Examples

			More generally, if Sum_{n>=0} log(1 + b*p^n*x + d*q^n*x^2)^n/n! = Sum_{n>=0} a(n)*x^n then a(n) = Sum_{k=0..[n/2]} C(n-k,k)*b^(n-2k)*d^k*C(p^(n-2k)*q^k,n-k).
		

Crossrefs

Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136637 (row sums).

Programs

Mathematica

Table[Sum[Binomial[n-k,k]*Binomial[2^k*3^(n-2*k),n-k], {k, 0, Floor[n/2]}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

PARI

{a(n)=sum(k=0,n\2,binomial(n-k,k)*binomial(3^(n-2*k)*2^k,n-k))}

PARI

/* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+3^i*x+2^i*x^2)^i/i!),n,x)}

Formula

G.f.: A(x) = Sum_{n>=0} log(1 + 3^n*x + 2^n*x^2)^n / n!.

a(n) ~ 3^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016