%I #23 Apr 11 2017 07:12:52

%S 2,24,386,6832,128442,2505720,50153770,1022997344,21170657906,

%T 443175051304,9363994959442,199387678947184,4273249614310458,

%U 92093491647488488,1994264443643492586,43366549723376465600,946516448918441722706,20726326157010810068856

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

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

%H Vaclav Kotesovec, <a href="/A219431/b219431.txt">Table of n, a(n) for n = 1..730</a> (first 180 terms from Paul D. Hanna)

%F a(n) ~ c * exp(Pi*n) / n, where c = 0.107862... . - _Vaclav Kotesovec_, Nov 29 2015

%e 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 +...

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

%e 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 +...

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

%t 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 *)

%t 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 *)

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

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

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

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

%Y Cf. A219430, A015128.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Nov 19 2012