A331725 E.g.f.: exp(x/(1 - x)) / (1 + x).

1, 0, 3, 4, 57, 216, 2755, 18348, 247569, 2368432, 35256771, 436248660, 7235178313, 108919083144, 2010150360387, 35421547781116, 723689454172065, 14543895730321248, 326843345169621379, 7354350135365751972, 180610925178770615001, 4488323611011676811320

Offset: 0

Author

Ilya Gutkovskiy, Jan 25 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..444

Crossrefs

Cf. A000262, A002720, A009940, A133942, A182386, A206703.

Programs

Maple

A331725 := proc(n)
    add((-1)^k*binomial(n,k)*k!*A000262(n-k),k=0..n) ;
end proc:
seq(A331725(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

Mathematica

nmax = 21; CoefficientList[Series[Exp[x/(1 - x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
A000262[n_] := If[n == 0, 1, n! Sum[Binomial[n - 1, k]/(k + 1)!, {k, 0, n - 1}]]; a[n_] := Sum[(-1)^k Binomial[n, k] k! A000262[n - k], {k, 0, n}]; Table[a[n], {n, 0, 21}]
a[n_] := (-1)^n n! (1 - Sum[(-1)^j*LaguerreL[j, 1, -1]/(j+1), {j,0,n-1}]);
Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 20 2022 *)

PARI

seq(n)={Vec(serlaplace(exp(x/(1 - x) + O(x*x^n)) / (1 + x)))} \\ Andrew Howroyd, Jan 25 2020

SageMath

def gen_a():
    F, L, S, N = 1, 1, 1, 1
    while True:
        yield F * S
        L = gen_laguerre(N - 1, 1, -1) / N
        S += L if F < 0 else -L
        F *= -N; N += 1
a = gen_a(); print([next(a) for  in range(21)]) # _Peter Luschny, Feb 20 2022

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * k! * A000262(n-k).

a(n) ~ n^(n - 1/4) / (2^(3/2) * exp(1/2 - 2*sqrt(n) + n)). - Vaclav Kotesovec, Jan 26 2020

D-finite with recurrence a(n) +(-n+1)*a(n-1) -(n-1)*(n+1)*a(n-2) +(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Aug 20 2021

a(n) = Sum_{k=0..n} (-1)^k * A206703(n,k). - Alois P. Heinz, Feb 19 2022

a(n) = (-1)^n*n!*(1 - Sum_{j=0..n-1}((-1)^j*LaguerreL(j, 1, -1)/(j + 1))). - Peter Luschny, Feb 20 2022