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A302870 Expansion of e.g.f. 1 / Sum_{n >= 0} (n+1)^3*x^n/n!.

%I A302870 #30 Mar 08 2024 12:00:27
%S A302870 1,-8,101,-1840,44441,-1340696,48530653,-2049479216,98915010545,
%T A302870 -5370730092136,324012625790741,-21502216185516848,
%U A302870 1556657523678767881,-122085765970981019000,10311495889448094131981,-933128678308256836233136,90072066063382006331898593
%N A302870 Expansion of e.g.f. 1 / Sum_{n >= 0} (n+1)^3*x^n/n!.
%C A302870 From _Vaclav Kotesovec_, Apr 15 2018: (Start)
%C A302870 In general, for m>=0, Sum_{k>=0} (k+1)^m * x^k / k! = exp(x) * Sum_{j = 1..m+1} Stirling2(m+1, j) * x^(j-1).
%C A302870 If m tends to infinity, then the real root of the equation Sum_{j = 1..m+1} Stirling2(m + 1, j) * x^(j-1) = 0, with a minimal absolute value, tends to -1/2^m.
%C A302870 (End)
%H A302870 Seiichi Manyama, <a href="/A302870/b302870.txt">Table of n, a(n) for n = 0..345</a>
%F A302870 From _Vaclav Kotesovec_, Apr 15 2018: (Start)
%F A302870 E.g.f: exp(-x)/(1 + 7*x + 6*x^2 + x^3).
%F A302870 a(n) ~ (-1)^n * n! * (371 + 414*r + 74*r^2) * exp(-r) * (7 + 6*r + r^2)^n / 257, where r = -0.16575681568607828288437387419... is the real root of the equation 1 + 7*r + 6*r^2 + r^3 = 0.
%F A302870 (End)
%t A302870 nmax = 20; CoefficientList[Series[1/(E^x*((1 + 7*x + 6*x^2 + x^3))), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 15 2018 *)
%Y A302870 Cf. A302189.
%K A302870 sign
%O A302870 0,2
%A A302870 _Seiichi Manyama_, Apr 15 2018