A369751 Expansion of e.g.f. exp(1 - (1+x)^3).

1, -3, 3, 21, -63, -423, 1899, 15201, -72063, -832491, 3105459, 60090093, -110508543, -5224722831, -3828328677, 510699368313, 2104026859521, -52582823289171, -473592954347037, 5168227121231301, 92434892126557761, -357595962971807223, -17085974691782295477

Offset: 0

Author

Seiichi Manyama, Jan 30 2024

Keywords

• sign

Crossrefs

Column k=3 of A369738.

Cf. A000587, A192989.

Programs

Maple

A369751 := proc(n)
    option remember ;
    if n =0 then
        1;
    else
        add( binomial(2,k-1) * procname(n-k)/(n-k)!,k=1..min(3,n)) ;
        -3*(n-1)!*% ;
    end if;
end proc:
seq(A369751(n),n=0..20) ; # R. J. Mathar, Feb 02 2024

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1-(1+x)^3)))

Formula

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

a(n) = Sum_{k=0..n} 3^k * Stirling1(n,k) * A000587(k).

D-finite with recurrence a(n) +3*a(n-1) +6*(n-1)*a(n-2) +3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Feb 02 2024