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A340904 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_1(k) * a(n-k).

%I A340904 #13 Mar 29 2022 15:11:12
%S A340904 1,1,5,28,225,2206,26174,361278,5704401,101297701,1998893240,
%T A340904 43386854622,1027353587730,26353742447280,728030940612638,
%U A340904 21548668265211778,680330296613877761,22821706122361385354,810587673640374442445,30390159250481750866640
%N A340904 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_1(k) * a(n-k).
%H A340904 Seiichi Manyama, <a href="/A340904/b340904.txt">Table of n, a(n) for n = 0..404</a>
%F A340904 E.g.f.: 1 / (1 - Sum_{i>=1} Sum_{j>=1} i * x^(i*j) / (i*j)!).
%F A340904 E.g.f.: 1 / (1 - Sum_{k>=1} sigma_1(k) * x^k/k!). - _Seiichi Manyama_, Mar 29 2022
%t A340904 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] DivisorSigma[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%t A340904 nmax = 19; CoefficientList[Series[1/(1 - Sum[Sum[i x^(i j)/(i j)!, {j, 1, nmax}], {i, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%o A340904 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, sigma(k)*x^k/k!)))) \\ _Seiichi Manyama_, Mar 29 2022
%o A340904 (PARI) a(n) = if(n==0, 1, sum(k=1, n, sigma(k)*binomial(n, k)*a(n-k))); \\ _Seiichi Manyama_, Mar 29 2022
%Y A340904 Cf. A000203, A006153, A180305, A274804, A340903.
%K A340904 nonn
%O A340904 0,3
%A A340904 _Ilya Gutkovskiy_, Jan 26 2021