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Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k).
3

%I #20 Aug 16 2022 07:43:58

%S 1,1,11,199,7585,427961,37901851,4526311231,729098029409,

%T 149311985624785,38243144308952971,11913301283967428951,

%U 4445712423354285230401,1954806416110914007773769,1000799932457357582959443035,589931632494798210345741193231

%N Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k).

%H Seiichi Manyama, <a href="/A352842/b352842.txt">Table of n, a(n) for n = 0..232</a>

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * sigma_k(k) * a(n-k)/(n-k)!.

%F a(n) ~ n! * n^n. - _Vaclav Kotesovec_, Apr 15 2022

%t nmax = 20; CoefficientList[Series[E^(Sum[DivisorSigma[k, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 15 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sigma(k, k)*x^k))))

%o (PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sigma(k, k)*a(n-k)/(n-k)!));

%Y Cf. A294363, A294361, A294362.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 05 2022