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

1, 0, -1, 0, 2, -3, -4, 30, -55, -126, 1190, -3333, -4522, 90354, -417349, 290628, 9897158, -79200003, 253387112, 981776010, -18675127711, 120607895430, -211057884214, -4240412978637, 54534511617962, -318859368844866, -1539810740197, 21691972838975040

Offset: 0

Author

Ilya Gutkovskiy, Jul 10 2020

Keywords

• sign

Crossrefs

Cf. A000587, A014619, A032347.

Programs

Mathematica

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]
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]!
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}]

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)}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Jul 10 2020

Formula

E.g.f. A(x) satisfies: A(x) = 1 - x + Integral (exp(-x) * A(x)) dx.

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