OFFSET
0,4
COMMENTS
Stirling transform of A000240.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..424
FORMULA
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000240(k).
a(n) ~ n! * exp(-1) / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020
MAPLE
g:= proc(n) option remember;
`if`(n=0, 0, n*(g(n-1)-(-1)^n))
end:
b:= proc(n, m) option remember; `if`(n=0,
g(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..20); # Alois P. Heinz, Jun 23 2023
MATHEMATICA
nmax = 20; CoefficientList[Series[(Exp[x] - 1) Exp[1 - Exp[x]]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
A000240[n_] := n! Sum[(-1)^k/k!, {k, 0, n - 1}]; a[n_] := Sum[StirlingS2[n, k] A000240[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, -k, 0] BellB[n - k, -1], {k, 1, n}], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 26 2020
STATUS
approved