A368766 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+1,2) / k!).

1, 0, 3, 3, 22, 95, 591, 4109, 32908, 296127, 2961325, 32574509, 390894186, 5081624327, 71142740683, 1067141110125, 17074257762136, 290262381956159, 5224722875211033, 99269734629009437, 1985394692580188950, 41693288544183967719, 917252347972047290071

Offset: 0

Author

Seiichi Manyama, Jan 04 2024

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A368765, A368767, A368768.

Cf. A009574, A368762.

Programs

Mathematica

nxt[{n_,a_}]:={n+1,a(n+1)+(-1)^(n+1) Binomial[n+2,2]}; NestList[nxt,{0,1},30][[;;,2]] (* Harvey P. Dale, Mar 26 2025 *)

PARI

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

Formula

a(0) = 1; a(n) = n*a(n-1) + (-1)^n * binomial(n+1,2).

a(n) = n! + (-1)^n * A009574(n).

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