A167141 G.f.: Sum_{n>=0} A004123(n)^2*log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)*log(1+x)^n/n!.

1, 4, 48, 864, 20880, 632448, 23018688, 978179328, 47529084096, 2598928566336, 157937795847936, 10559489876375040, 770269715428025088, 60876094422772800000, 5181654464327251948032, 472584847824904789910016

Offset: 0

Author

Paul D. Hanna, Nov 04 2009

Keywords

• nonn

Comments

CONJECTURE: For all integer 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.

In this case, m=2 and L(n) = A004123(n), which is the number of generalized weak orders on n points.

Examples

			G.f.: A(x) = 1 + 4*x + 48*x^2 + 864*x^3 + 20880*x^4 + 632448*x^5 +...
Illustrate A(x) = Sum_{n>=0} A004123(n)^2 * log(1+x)^n/n!:
A(x) = 1 + 2^2*log(1+x) + 10^2*log(1+x)^2/2! + 74^2*log(1+x)^3/3! + 730^2*log(1+x)^4/4! + 9002^2*log(1+x)^5/5! +...+ A004123(n)^2*log(1+x)^n/n! +...
where the e.g.f. of A004123 is 1/(3 - 2*exp(x)) and thus:
1/(1-2x) = 1 + 2*log(1+x) + 10*log(1+x)^2/2! + 74*log(1+x)^3/3! + 730*log(1+x)^4/4! + 9002*log(1+x)^5/5! +...+ A004123(n)*log(1+x)^n/n! +...
		

Crossrefs

Cf. A004123, variants: A167139, A167138, A101370.

Programs

PARI

{A004123(n)=sum(k=0,n,2^k*stirling(n, k, 2)*k!)}
{a(n)=polcoeff(sum(m=0,n,A004123(m)^2*log(1+x+x*O(x^n))^m/m!),n)}