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A303004 Expansion of e.g.f. exp(Sum_{k>=1} M(k)*x^k/k!), where M() is the exponential of Mangoldt function (A014963).

%I A303004 #5 Feb 16 2025 08:33:53
%S A303004 1,1,3,10,39,186,962,5587,35367,241216,1771052,13827925,114558314,
%T A303004 1001769237,9208116647,88737108635,893505145271,9379190223746,
%U A303004 102402586369892,1160487000658679,13627075242031720,165524499516422471,2076762033563394443,26877177548737581587
%N A303004 Expansion of e.g.f. exp(Sum_{k>=1} M(k)*x^k/k!), where M() is the exponential of Mangoldt function (A014963).
%C A303004 Exponential transform of A014963.
%H A303004 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MangoldtFunction.html">Mangoldt Function</a>
%H A303004 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A303004 E.g.f.: exp(Sum_{k>=1} A014963(k)*x^k/k!).
%e A303004 E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 10*x^3/3! + 39*x^4/4! + 186*x^5/5! + 962*x^6/6! + 5587*x^7/7! + ...
%t A303004 nmax = 23; CoefficientList[Series[Exp[Sum[Exp[MangoldtLambda[k]] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A303004 a[n_] := a[n] = Sum[Exp[MangoldtLambda[k]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
%Y A303004 Cf. A014963, A140254, A140255.
%K A303004 nonn
%O A303004 0,3
%A A303004 _Ilya Gutkovskiy_, Apr 17 2018