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A292916 a(n) = n! * [x^n] exp(n*x)/(2 - exp(x)). 5
1, 2, 11, 94, 1083, 15666, 272451, 5532206, 128409707, 3352959850, 97259891163, 3102552150006, 107936130271899, 4066743353318114, 164961642651034547, 7167348523420169278, 332081754670735087275, 16343667009638859878298, 851478575825591156040843, 46814697307371602567813126 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The n-th term of the n-th binomial transform of A000670.
LINKS
N. J. A. Sloane, Transforms
FORMULA
a(n) = A292915(n,n).
a(n) ~ n! * 2^(n-1) / (log(2))^(n+1). - Vaclav Kotesovec, Sep 27 2017
a(n) = 2^n*A000670(n) - Sum_{k=0..n-1} 2^k*(n-1-k)^n. - Seiichi Manyama, Dec 25 2023
MAPLE
b:= proc(n, k) option remember; k^n +add(
binomial(n, j)*b(j, k), j=0..n-1)
end:
a:= n-> b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 27 2017
MATHEMATICA
Table[n! SeriesCoefficient[Exp[n x]/(2 - Exp[x]), {x, 0, n}], {n, 0, 19}]
Table[HurwitzLerchPhi[1/2, -n, n]/2, {n, 0, 19}]
PROG
(PARI) a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
a(n) = 2^n*a000670(n)-sum(k=0, n-1, 2^k*(n-1-k)^n); \\ Seiichi Manyama, Dec 25 2023
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 50);
A292916:= func< n | Coefficient(R!(Laplace( Exp(n*x)/(2-Exp(x)) )), n) >;
[A292916(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024
(SageMath) [factorial(n)*( exp(n*x)/(2-exp(x)) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024
CROSSREFS
Main diagonal of A292915.
Sequence in context: A083069 A158837 A236962 * A290586 A098621 A266834
KEYWORD
nonn,changed
AUTHOR
Ilya Gutkovskiy, Sep 26 2017
STATUS
approved

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Last modified June 15 09:26 EDT 2024. Contains 373407 sequences. (Running on oeis4.)