A371042 E.g.f. satisfies A(x) = 1 + x^2*exp(x*A(x)).

1, 0, 2, 6, 12, 140, 1470, 10122, 114296, 1874952, 25462170, 379431470, 7546461252, 151797222876, 3066316693622, 72101615826450, 1843378516587120, 47860832586054032, 1338908395558366386, 40675047500003794902, 1282380661224172506620

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k) * binomial(n-2*k+1,k)/( (n-2*k+1)*(n-2*k)! ).

From Vaclav Kotesovec, Mar 10 2024: (Start)

E.g.f.: 1 - LambertW(-exp(x)*x^3)/x.

a(n) ~ sqrt(1 + LambertW(exp(-1/3)/3)) * n^(n-1) / (exp(n) * 3^(n + 1/2) * LambertW(exp(-1/3)/3)^(n+1)). (End)

Mathematica

nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^3]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n-2*k+1, k)/((n-2*k+1)*(n-2*k)!));

Crossrefs

Cf. A370984, A371018, A371043.

Cf. A161631, A371044.

Cf. A364978.

Sequence in context: A325504 A193619 A290406 * A319481 A195338 A179201

Adjacent sequences: A371039 A371040 A371041 * A371043 A371044 A371045

Keyword

nonn

Author

Seiichi Manyama, Mar 09 2024

Status

approved