OFFSET
0,2
COMMENTS
From Vaclav Kotesovec, Apr 15 2018: (Start)
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).
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.
(End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..345
FORMULA
From Vaclav Kotesovec, Apr 15 2018: (Start)
E.g.f: exp(-x)/(1 + 7*x + 6*x^2 + x^3).
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.
(End)
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Apr 15 2018
STATUS
approved