A318224 a(n) = n! * [x^n] exp(x/(1 + n*x)).

1, 1, -3, 37, -1007, 47901, -3514499, 367671697, -51952729023, 9529552851193, -2201241933756899, 625136460673954461, -214066473170125310063, 86976878219664125966677, -41368038169392401671082787, 22767783580493235411255966601, -14356419990032448099044028030719

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = n! * [x^n] Product_{k>=1} exp((-n)^(k-1)*x^k).

a(n) = Sum_{k=0..n} (-n)^(n-k)*binomial(n-1,k-1)*n!/k!.

a(n) ~ -(-1)^n * c * n^(2*n - 1/2) / exp(n), where c = BesselJ(1,2) * sqrt(2*Pi) = 1.44563470980450699365002928132323794056211645203313522173628289... - Vaclav Kotesovec, Aug 21 2018

Mathematica

Table[n! SeriesCoefficient[Exp[x/(1 + n x)], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, n}], {n, 16}]]
Join[{1}, Table[(-1)^(n + 1) n^n (n - 1)! Hypergeometric1F1[1 - n, 2, 1/n], {n, 16}]]
Flatten[{1, Table[-(-1)^n * n^(n-1) * (n-1)! * LaguerreL[n-1, 1, 1/n], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 21 2018 *)

Crossrefs

Cf. A111884, A293146, A317279, A318223.

Sequence in context: A354020 A201697 A274308 * A300986 A003716 A331345

Adjacent sequences: A318221 A318222 A318223 * A318225 A318226 A318227

Keyword

sign

Author

Ilya Gutkovskiy, Aug 21 2018

Status

approved