A335595 E.g.f.: exp(-x * (2 + x)) / (1 - x)^2.

1, 0, 0, 4, 12, 48, 400, 3120, 25872, 251776, 2715264, 31809600, 405296320, 5580385536, 82469607168, 1302102360832, 21875297337600, 389590168842240, 7331376554610688, 145352459953603584, 3028176414606560256, 66135374473635328000, 1510938930307368898560, 36038691473858577444864

Offset: 0

Author

Ilya Gutkovskiy, Nov 20 2020

Keywords

• nonn

Crossrefs

Cf. A038205, A087981.

Programs

Maple

A335595 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        2*add(binomial(n-1,k-1)*(k-1)!*procname(n-k),k=3..n) ;
    end if;
end proc:
seq(A335595(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

Mathematica

nmax = 23; CoefficientList[Series[Exp[-x (2 + x)]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] HermiteH[k, -1] (n - k + 1)!, {k, 0, n}], {n, 0, 23}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(-x*(2+x))/(1-x)^2)) \\ Michel Marcus, Nov 21 2020

Formula

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

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

a(n) = Sum_{k=0..n} binomial(n,k) * A038205(k) * A038205(n-k).

a(n) ~ exp(-3) * n * n!. - Vaclav Kotesovec, Aug 09 2021

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