A219431 - OEIS

A219431

Logarithmic derivative of the g.f. of the overpartitions of n^2 (A219430).

2, 24, 386, 6832, 128442, 2505720, 50153770, 1022997344, 21170657906, 443175051304, 9363994959442, 199387678947184, 4273249614310458, 92093491647488488, 1994264443643492586, 43366549723376465600, 946516448918441722706, 20726326157010810068856

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OFFSET

1,1

COMMENTS

Conjecture: a(2*n+1) == 2 (mod 4), a(2*n) == 0 (mod 4).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..730 (first 180 terms from Paul D. Hanna)

FORMULA

a(n) ~ c * exp(Pi*n) / n, where c = 0.107862... . - Vaclav Kotesovec, Nov 29 2015

EXAMPLE

L.g.f.: L(x) = 2*x + 24*x^2/2 + 386*x^3/3 + 6832*x^4/4 + 128442*x^5/5 + 2505720*x^6/6 + 50153770*x^7/7 + 1022997344*x^8/8 +...

such that exponentiation yields the g.f. of A219430:

exp(L(x)) = 1 + 2*x + 14*x^2 + 154*x^3 + 2062*x^4 + 31066*x^5 + 504886*x^6 + 8652402*x^7 + 154208270*x^8 +...+ A219430(n)*x^n +...

where A219430(n) = A015128(n^2).

MATHEMATICA

nmax = 20; A015128 = Rest[CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n] - DivisorSigma[1, n])*(x^n/n), {n, 1, nmax^2}]], {x, 0, nmax^2}], x]]; A219430 = Table[A015128[[n^2]], {n, 1, nmax}]; Rest[CoefficientList[Series[Log[1 + Sum[A219430[[k]]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]] (* Vaclav Kotesovec, Nov 28 2015 *)

nmax = 20; Rest[CoefficientList[Series[Log[1 + Sum[Sum[PartitionsP[k^2 - j]*PartitionsQ[j], {j, 0, k^2}] * x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]] (* Vaclav Kotesovec, Nov 28 2015 *)

PROG

(PARI) /* From A219430(n) = [x^(n^2)] 1 / theta_4(x) */

{A219430(n)=polcoeff(1/(1+2*sum(k=1, n, (-x)^(k^2))+x*O(x^(n^2))), n^2)}

{a(n)=n*polcoeff(log(sum(k=0, n, A219430(k)*x^k)+x*O(x^n)), n)}

for(n=1, 25, print1(a(n), ", "))

CROSSREFS

Cf. A219430, A015128.

Sequence in context: A052670 A052736 A103904 * A214688 A364195 A003102

Adjacent sequences: A219428 A219429 A219430 * A219432 A219433 A219434

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 19 2012

STATUS

approved