A317363 Expansion of e.g.f. 1/(2 - exp(x/(1 + x))).

1, 1, 1, 1, 3, 1, 33, -83, 955, -5243, 44913, -285647, 1672179, 3544009, -352029311, 9470312053, -208005703605, 4326748972141, -88602638362863, 1819530461684473, -37722654765171965, 791428823931046321, -16784285106705759519, 358449656565896328061, -7653024671576463436197

Offset: 0

Author

Ilya Gutkovskiy, Jul 26 2018

Keywords

• sign

Comments

Inverse Lah transform of the Fubini numbers (A000670).

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000670, A084358.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= proc(n) option remember; add((-1)^(n-k)*
      n!/k!*binomial(n-1, k-1)*b(k), k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

Mathematica

nmax = 24; CoefficientList[Series[1/(2 - Exp[x/(1 + x)]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] HurwitzLerchPhi[1/2, -k, 0] n!/(2 k!), {k, 0, n}], {n, 0, 24}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*A000670(k)*n!/k!.