cp's OEIS Frontend

A318783 Expansion of Product_{k>=1} 1/(1 - x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

%I A318783 #7 Sep 03 2018 18:44:02
%S A318783 1,0,1,1,3,2,7,5,14,13,27,26,57,53,102,110,192,204,353,381,626,704,
%T A318783 1094,1246,1920,2185,3252,3800,5503,6440,9213,10827,15194,18035,24836,
%U A318783 29579,40369,48103,64758,77635,103279,123882,163506,196286,256688,308836,400329,481847,620832
%N A318783 Expansion of Product_{k>=1} 1/(1 - x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).
%C A318783 Convolution of A010815 and A006171.
%C A318783 Euler transform of A032741.
%H A318783 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A318783 G.f.: Product_{k>=1} 1/(1 - x^k)^A032741(k).
%F A318783 G.f.: exp(Sum_{k>=1} sigma_1(k)*x^(2*k)/(k*(1 - x^k))), where sigma_1(k) = sum of divisors of k (A000203).
%p A318783 with(numtheory):
%p A318783 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A318783       (tau(d)-1), d=divisors(j))*a(n-j), j=1..n)/n)
%p A318783     end:
%p A318783 seq(a(n), n=0..50);  # _Alois P. Heinz_, Sep 03 2018
%t A318783 nmax = 48; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
%t A318783 nmax = 48; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A318783 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]
%Y A318783 Cf. A000005, A000203, A006171, A010815, A032741, A318784.
%K A318783 nonn
%O A318783 0,5
%A A318783 _Ilya Gutkovskiy_, Sep 03 2018