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a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_1(k) * a(n-k).
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%I #13 Mar 29 2022 15:11:12

%S 1,1,5,28,225,2206,26174,361278,5704401,101297701,1998893240,

%T 43386854622,1027353587730,26353742447280,728030940612638,

%U 21548668265211778,680330296613877761,22821706122361385354,810587673640374442445,30390159250481750866640

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

%H Seiichi Manyama, <a href="/A340904/b340904.txt">Table of n, a(n) for n = 0..404</a>

%F E.g.f.: 1 / (1 - Sum_{i>=1} Sum_{j>=1} i * x^(i*j) / (i*j)!).

%F E.g.f.: 1 / (1 - Sum_{k>=1} sigma_1(k) * x^k/k!). - _Seiichi Manyama_, Mar 29 2022

%t 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 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 (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 (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 Cf. A000203, A006153, A180305, A274804, A340903.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jan 26 2021