A167540 G.f.: Sum_{n>=0} A155585(2n+1)*log(1-2x)^n/n!, where (1-2*x)^2/(1-2*x+2*x^2) = Sum_{n>=0} A155585(n)*log(1-2x)^n/n!.

1, 4, 36, 432, 6504, 118272, 2525824, 62011648, 1721422656, 53324108032, 1823657963776, 68252530738176, 2774853481548288, 121780933815238656, 5738394351838543872, 288958047466769973248, 15485497781445500923904

Offset

0,2

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

Links

Table of n, a(n) for n=0..16.

Example

G.f.: A(x) = 1 + 4*x + 36*x^2 + 432*x^3 + 6504*x^4 + 118272*x^5 +...
Illustrate the g.f.:
A(x) = 1 - 2*log(1-2*x) + 16*log(1-2*x)^2/2! - 272*log(1-2*x)^3/3! + 7936*log(1-2*x)^4/4! - 353792*log(1-2*x)^5/5! +...+ A155585(2n+1)*log(1-2x)^n/n! +...
where:
(1-2*x)^2/(1-2*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! +...+ A155585(n)*log(1-2x)^n/n! +...

Prog

(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(2*k+1)*log(1-2*x +x*O(x^n))^k/k!), n)}

Crossrefs

Cf. A155585, variant: A167532.

Sequence in context: A132864 A294050 A052700 * A136224 A321963 A307903

Adjacent sequences: A167537 A167538 A167539 * A167541 A167542 A167543

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2009

Status

approved