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A326829 G.f. A(x) satisfies: A(x) = 1 - x * (1 - A(x/(1 + x)) / (1 + x)).

%I A326829 #75 Jan 11 2021 23:38:14
%S A326829 1,0,-1,0,2,-3,-4,30,-55,-126,1190,-3333,-4522,90354,-417349,290628,
%T A326829 9897158,-79200003,253387112,981776010,-18675127711,120607895430,
%U A326829 -211057884214,-4240412978637,54534511617962,-318859368844866,-1539810740197,21691972838975040
%N A326829 G.f. A(x) satisfies: A(x) = 1 - x * (1 - A(x/(1 + x)) / (1 + x)).
%F A326829 E.g.f. A(x) satisfies: A(x) = 1 - x + Integral (exp(-x) * A(x)) dx.
%F A326829 a(0) = 1, a(1) = 0; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n-1,k-1) * a(n-k).
%t A326829 nmax = 27; A[_] = 0; Do[A[x_] = 1 - x (1 - A[x/(1 + x)]/(1 + x)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A326829 nmax = 27; A[_] = 0; Do[A[x_] = 1 - x + Integrate[Exp[-x] A[x + O[x]^(nmax + 1)], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
%t A326829 a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
%o A326829 (PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1 - x * (1 - subst(A, x, x/(1+x)) / (1+x))); polcoeff(A, n)}
%o A326829 for(n=0, 30, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Jul 10 2020
%Y A326829 Cf. A000587, A014619, A032347.
%K A326829 sign
%O A326829 0,5
%A A326829 _Ilya Gutkovskiy_, Jul 10 2020