A344398 a(n) = (-1)^n * F_{n}((-1)^n * n), where F_{n}(x) is the Fubini polynomial.

1, 1, 10, 111, 8676, 243005, 49729758, 2634606331, 1026912225160, 88276603008249, 55954905981282210, 7103694104486331671, 6655958151527584785900, 1171100778886715057133493, 1521436331153097968932487206, 354408430829377435361459172915, 609729139653483641913607434550800

Offset: 0

Author

Peter Luschny, May 21 2021

Keywords

• nonn

Crossrefs

The coefficients of the Fubini polynomials are A131689.

Cf. A094420.

Programs

Maple

F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
a := n -> (-1)^n*subs(x = (-1)^n*n, F(n)):
seq(a(n), n = 0..17);

Mathematica

F[n_][x_] := If[n == 0, 1, Sum[k! StirlingS2[n, k] x^k, {k, 0, n}]];
a[n_] := (-1)^n F[n][(-1)^n*n];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, May 09 2024 *)

SageMath

@cached_function
def F(n):
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
def a(n):
    return (-1)^n*F(n).substitute(x = (-1)^n*n)
print([a(n) for n in range(17)])