A167138 G.f.: Sum_{n>=0} A167137(n)^2 * log(1+x)^n/n! where Sum_{n>=0} A167137(n)*log(1+x)^n/n! = g.f. of the partition numbers (A000041).

1, 1, 12, 148, 2523, 48996, 1127354, 29348080, 849632392, 27096593838, 943340417806, 35501579861404, 1434531966551084, 61939404662074706, 2844544965703554566, 138338597978951126666, 7098617731036257970895

Offset: 0

Author

Paul D. Hanna, Nov 03 2009

Keywords

• nonn

Comments

Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

Examples

			G.f.: A(x) = 1 + x + 12*x^2 + 148*x^3 + 2523*x^4 + ...
Illustrate A(x) = Sum_{n>=0} A167137(n)^2*log(1+x)^n/n!:
A(x) = 1 + log(1+x) + 5^2*log(1+x)^2/2! + 31^2*log(1+x)^3/3! + 257^2*log(1+x)^4/4! + ...
where P(x), the partition function of A000041, is generated by:
P(x) = 1 + log(1+x) + 5*log(1+x)^2/2! + 31*log(1+x)^3/3! + 257*log(1+x)^4/4! + ...
		

Crossrefs

Cf. A167137, A000041.

Programs

PARI

{A167137(n)=sum(k=0,n,numbpart(k)*stirling(n, k, 2)*k!)}
{a(n)=polcoef(sum(m=0,n,A167137(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

Formula

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A167137(k)^2. - Vladeta Jovovic, Nov 08 2009