A347978 - OEIS

%I #11 Sep 12 2025 11:00:25

%S 1,-1,0,-3,4,-30,186,-630,11600,-26712,1005480,-2581920,117196872,

%T -485308824,17734457664,-131070696120,3387342915840,-43890398953920,

%U 801577841697216,-17363169328243392,233460174245351040,-7968629225100337920,84363134551361043840

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

%H Bernd C. Kellner, <a href="https://arxiv.org/abs/2509.05235">Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients</a>, arXiv:2509.05235 [math.NT], 2025. See p. 11.

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

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

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

%t nmax = 22; CoefficientList[Series[1/(1 + x)^(1/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

%t 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}]

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(1/(1+x)^(1/(1-x)))) \\ _Michel Marcus_, Sep 22 2021

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

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Sep 22 2021