# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a219431 Showing 1-1 of 1 %I A219431 #23 Apr 11 2017 07:12:52 %S A219431 2,24,386,6832,128442,2505720,50153770,1022997344,21170657906, %T A219431 443175051304,9363994959442,199387678947184,4273249614310458, %U A219431 92093491647488488,1994264443643492586,43366549723376465600,946516448918441722706,20726326157010810068856 %N A219431 Logarithmic derivative of the g.f. of the overpartitions of n^2 (A219430). %C A219431 Conjecture: a(2*n+1) == 2 (mod 4), a(2*n) == 0 (mod 4). %H A219431 Vaclav Kotesovec, Table of n, a(n) for n = 1..730 (first 180 terms from Paul D. Hanna) %F A219431 a(n) ~ c * exp(Pi*n) / n, where c = 0.107862... . - _Vaclav Kotesovec_, Nov 29 2015 %e A219431 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 A219431 such that exponentiation yields the g.f. of A219430: %e A219431 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 A219431 where A219430(n) = A015128(n^2). %t A219431 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 A219431 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 A219431 (PARI) /* From A219430(n) = [x^(n^2)] 1 / theta_4(x) */ %o A219431 {A219430(n)=polcoeff(1/(1+2*sum(k=1,n,(-x)^(k^2))+x*O(x^(n^2))),n^2)} %o A219431 {a(n)=n*polcoeff(log(sum(k=0,n,A219430(k)*x^k)+x*O(x^n)),n)} %o A219431 for(n=1,25,print1(a(n),", ")) %Y A219431 Cf. A219430, A015128. %K A219431 nonn %O A219431 1,1 %A A219431 _Paul D. Hanna_, Nov 19 2012 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE