|
%I #9 Apr 03 2019 03:00:15
%S 1,1,3,7,11,25,51,87,129,286,462,760,1312,2102,3470,5988,8840,13884,
%T 22577,33545,55961,85341,126705,194317,293621,435040,641472,971503,
%U 1462483,2108161,3124489,4474579,6545809,9561923,13518678,19809034,28387625,40286631,57039233
%N Expansion of Product_{k>=1} (1 + sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).
%H Vaclav Kotesovec, <a href="/A316962/b316962.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*sigma(j)^k*x^(j*k)/k).
%p with(numtheory): a:=series(mul(1+sigma(k)*x^k,k=1..100),x=0,39): seq(coeff(a,x,n),n=0..38); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 38; CoefficientList[Series[Product[(1 + DivisorSigma[1, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 38; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) DivisorSigma[1, j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSigma[1, d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]
%Y Cf. A000203, A180305, A192065, A279786, A316961.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 17 2018
|
