The On-Line Encyclopedia of Integer Sequences (OEIS)

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A182818

G.f.: exp( Sum_{n>=1} sigma(2n)*x^n/n ).
(history; published version)

#39 by Michel Marcus at Wed Jan 25 14:39:05 EST 2017

STATUS

editing

proposed

#38 by Michel Marcus at Wed Jan 25 14:39:00 EST 2017

PROG

(PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)^3) \\ Joerg Arndt, Dec 05 2010]

STATUS

proposed

editing

#37 by Peter Bala at Wed Jan 25 14:16:26 EST 2017

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editing

proposed

#36 by Peter Bala at Tue Jan 24 11:05:50 EST 2017

FORMULA

G.f.: A(x) = E(x^2)/E(x)^3 where E(x)=prod(n>=1, 1-x^n). [Joerg Arndt, Dec 05 2010]

The ordinary generating function A(x) is the infinite product AF(x) * AF(x^2) * AF(x^3) * ..., where AF(x) is the ordinary generating function of A005408. - Gary W. Adamson, Jul 15 2012

From Peter Bala, Jan 24 2016: (Start)

a(n) = Sum_{k = 0..2*n} (-1)^k*p(k)*p(2*n-k), where p(n) = A000041(n) is the partition function.

A(x^2) = 1/Product_{n>=1} (1 - (-x)^n) * 1/Product_{n>=1} (1 - x^n). (End)

MAPLE

with(combinat):

seq(add((-1)^k*numbpart(k)*numbpart(2*n - k), k = 0..2*n), n = 0..40);

KEYWORD

nonn,easy

STATUS

approved

editing

#35 by Vaclav Kotesovec at Sun Nov 29 03:49:36 EST 2015

STATUS

editing

approved

#34 by Vaclav Kotesovec at Sun Nov 29 03:15:31 EST 2015

FORMULA

a(n) ~ 5^(3/4) * exp(Pi*sqrt(5*n/3)) / (16 * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Nov 29 2015

#33 by Vaclav Kotesovec at Sun Nov 29 03:10:46 EST 2015

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[(1+x^k)/(1-x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

#32 by Vaclav Kotesovec at Sun Nov 29 03:07:38 EST 2015

MATHEMATICA

nmax = 40; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n])*(x^n/n), {n, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

STATUS

approved

editing

#31 by Vaclav Kotesovec at Sun Nov 29 03:03:53 EST 2015

STATUS

editing

approved

#30 by Vaclav Kotesovec at Sun Nov 29 03:03:36 EST 2015

LINKS

Vaclav Kotesovec, <a href="/A182818/b182818.txt">Table of n, a(n) for n = 0..10000</a>

STATUS

approved

editing