A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Formula

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.

(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).

From Seiichi Manyama, Jul 09 2022: (Start)

a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.

a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

Mathematica

CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!

Prog

(PARI) x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017
(PARI) a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

Crossrefs

Column k=2 of A292892.

Cf. A000110, A052506, A216507, A060311.

Cf. A216688, A216689, A080108, A000248.

Sequence in context: A303481 A247256 A292951 * A366564 A247212 A358013

Adjacent sequences: A240986 A240987 A240988 * A240990 A240991 A240992

Keyword

nonn

Author

Vaclav Kotesovec, Aug 06 2014

Status

approved