A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(n - 3*k)!.

1, 1, 1, 1, 25, 121, 361, 5881, 82321, 547345, 6053041, 167991121, 2179469161, 22892967241, 788375451865, 18046198202761, 245523704069281, 7548055281543841, 270833271588545761, 5369819950838359585, 141456920470310708281, 6760255576117937586841

Offset: 0

Author

Seiichi Manyama, Aug 18 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..393

Crossrefs

Cf. A354436, A356628, A356630.

Cf. A354553, A356633, A358065.

Programs

Mathematica

a[n_] := n! * Sum[(n - 3*k)^k/(n - 3*k)!, {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)

PARI

a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(n-3*k)!);

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^3)))))

Formula

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^3)).

a(n) ~ sqrt(Pi/3) * exp((2*n - 3)/(6*LambertW(exp(1/4)*(2*n - 3)/8)) - 4*n/3) * n^(4*n/3 + 1/2) / (sqrt(1 + LambertW(exp(1/4)*(2*n - 3)/8)) * 2^(2*n/3 + 1/2) * LambertW(exp(1/4)*(2*n - 3)/8)^(n/3)). - Vaclav Kotesovec, Nov 01 2022