A167532 -id:A167532 - cp's OEIS frontend

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

Paul D. Hanna, Nov 06 2009

nonn

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.

			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! +...

Cf. A155585, variant: A167532.

{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)}

cp's OEIS Frontend

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

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cp's OEIS Frontend

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!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

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