cp's OEIS Frontend

A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

%I A319107 #8 Apr 03 2019 02:55:27
%S A319107 1,0,1,1,3,2,9,5,17,17,38,33,88,75,169,181,343,353,712,728,1348,1518,
%T A319107 2591,2898,5025,5615,9259,10866,17160,20111,31775,37264,57130,68782,
%U A319107 102663,123698,183793,221708,323077,395325,566079,693248,987086,1210110,1700074,2100674,2915549
%N A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).
%C A319107 Convolution of A192065 and A255528.
%C A319107 Weigh transform of A001065.
%H A319107 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A319107 G.f.: Product_{k>=1} (1 + x^k)^A001065(k).
%F A319107 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*(sigma_1(d) - d) ) * x^k/k).
%F A319107 a(n) ~ exp(-Pi^4 / (864*(Pi^2 - 6)*Zeta(3)) - Pi^2 * n^(1/3) / (12*(2*(Pi^2 - 6)*Zeta(3))^(1/3)) + 3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (2^(17/24) * sqrt(3*Pi) * n^(2/3)). - _Vaclav Kotesovec_, Sep 11 2018
%p A319107 with(numtheory): a:=series(mul((1+x^k)^(sigma(k)-k),k=1..100),x=0,47): seq(coeff(a,x,n),n=0..46); # _Paolo P. Lava_, Apr 02 2019
%t A319107 nmax = 46; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A319107 nmax = 46; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
%t A319107 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]
%Y A319107 Cf. A000203, A001065, A192065, A255528, A318784.
%K A319107 nonn
%O A319107 0,5
%A A319107 _Ilya Gutkovskiy_, Sep 10 2018