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A256016
a(n) = n! * Sum_{k=0..n} k^n/k!.
18
1, 1, 6, 57, 796, 15145, 374526, 11669665, 447595800, 20733553809, 1141067915290, 73552752257281, 5484203261135028, 467864288815609465, 45236104846954021014, 4915818294874879570305, 596044703812665607374256, 80118478395137652912476449, 11870487496575403846760198322
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Bell Polynomial.
FORMULA
a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.
a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).
E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - Seiichi Manyama, Aug 23 2022
a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024
MATHEMATICA
Join[{1}, Table[n!*Sum[k^n/k!, {k, 0, n}], {n, 1, 20}]]
PROG
(PARI) a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-k*x))))) \\ Seiichi Manyama, Aug 23 2022
CROSSREFS
Main diagonal of A337085.
Sequence in context: A242817 A376100 A295238 * A361291 A145170 A180255
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jun 01 2015
EXTENSIONS
a(0)=1 prepended by Seiichi Manyama, Aug 14 2020
STATUS
approved