A347978 Expansion of e.g.f. 1/(1 + x)^(1/(1 - x)).

1, -1, 0, -3, 4, -30, 186, -630, 11600, -26712, 1005480, -2581920, 117196872, -485308824, 17734457664, -131070696120, 3387342915840, -43890398953920, 801577841697216, -17363169328243392, 233460174245351040, -7968629225100337920, 84363134551361043840

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Bernd C. Kellner, Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients, arXiv:2509.05235 [math.NT], 2025. See p. 11.

Formula

E.g.f.: exp( Sum_{k>=1} x^k * Sum_{j=1..k} (-1)^j / j ).

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * A024167(k) * a(n-k).

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * A073478(k) * a(n-k).

Mathematica

nmax = 22; CoefficientList[Series[1/(1 + x)^(1/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
A024167[n_] := n! Sum[(-1)^(k + 1)/k, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] A024167[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(1/(1+x)^(1/(1-x)))) \\ Michel Marcus, Sep 22 2021

Crossrefs

Cf. A005727, A007120, A008405, A024167, A058312, A058313, A073478, A087761.

Sequence in context: A225629 A032833 A151466 * A127068 A372794 A058603

Adjacent sequences: A347975 A347976 A347977 * A347979 A347980 A347981

Keyword

sign

Author

Ilya Gutkovskiy, Sep 22 2021

Status

approved