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Revision History for A167532 (Underlined text is an addition; strikethrough text is a deletion.)

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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!.
(history; published version)
#2 by Russ Cox at Fri Mar 30 18:37:20 EDT 2012
AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Nov 05 2009

Discussion
Fri Mar 30 18:37
OEIS Server: https://oeis.org/edit/global/213
#1 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010
NAME

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!.

DATA

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

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 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.

EXAMPLE

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! +...

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 05 2009

STATUS

approved

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Last modified August 30 21:20 EDT 2024. Contains 375548 sequences. (Running on oeis4.)