%I #30 Mar 08 2024 12:00:27

%S 1,-8,101,-1840,44441,-1340696,48530653,-2049479216,98915010545,

%T -5370730092136,324012625790741,-21502216185516848,

%U 1556657523678767881,-122085765970981019000,10311495889448094131981,-933128678308256836233136,90072066063382006331898593

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

%C From _Vaclav Kotesovec_, Apr 15 2018: (Start)

%C 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 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 (End)

%H Seiichi Manyama, <a href="/A302870/b302870.txt">Table of n, a(n) for n = 0..345</a>

%F From _Vaclav Kotesovec_, Apr 15 2018: (Start)

%F E.g.f: exp(-x)/(1 + 7*x + 6*x^2 + x^3).

%F 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 (End)

%t 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 Cf. A302189.

%K sign

%O 0,2

%A _Seiichi Manyama_, Apr 15 2018