A167532 G.f.: Sum_{n>=0} A155585(n)^2 * log(1/(1-2*x))^n/n!, where 1/(1-2*x+2*x^2) = Sum_{n>=0} A155585(n)*log(1/(1-2*x))^n/n!.

1, 2, 2, 8, 20, 112, 432, 3200, 16704, 154688, 1017920, 11333888, 90011264, 1172330496, 10908526592, 162802835456, 1737036006400, 29235365490688, 351847501606912, 6593866787569664, 88364197074231296, 1825016315965767680

Offset: 0

Author

Paul D. Hanna, Nov 05 2009

Keywords

• nonn

Comments

Note that A155585(n) = 2^n E_{n}(1) where E_{n}(x) are the Euler polynomials; e.g.f. of A155585 is exp(x)/cosh(x).

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.

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 8*x^3 + 20*x^4 + 112*x^5 + 432*x^6 +...
Illustrate A(x) = Sum_{n>=0} A155585(n)^2*log(1/(1-2*x))^n/n!:
A(x) = 1 - log(1-2*x) - 2^2*log(1-2*x)^3/3! - 16^2*log(1-2*x)^5/5! - 272^2*log(1-2*x)^7/7! - 7936^2*log(1-2*x)^9/9! +...+ A155585(n)^2*[ -log(1-2x)]^n/n! +...
where:
1/((1-x)^2 + x^2) = 1 - log(1-2*x) + 2*log(1-2*x)^3/3! - 16*log(1-2*x)^5/5! + 272*log(1-2*x)^7/7! - 7936*log(1-2*x)^9/9! +...+ A155585(n)*[ -log(1-2x)]^n/n! +...
		

Crossrefs

Cf. A155585, variants: A167141, A167139, A167138, A101370.

Programs

PARI

{A155585(n)=if(n==0,1,bernfrac(n+1)*(2^(n+1)-1)*2^(n+1)/(n+1))}
{a(n)=polcoeff(sum(k=0,n,A155585(k)^2*log(1/(1-2*x +x*O(x^n)))^k/k!),n)}