A307355 E.g.f. A(x) satisfies: d/dx A(x) = 1 + A(x/(1 + x)).

1, 1, -1, -1, 23, -197, 1093, 5377, -374863, 9934889, -195976201, 2134159519, 67270069831, -6730482201869, 365726633654957, -15494292929032063, 458469986808144737, 2828723973314497873, -2067360599320208561297, 238687490584532161293631, -20425982116564721266720009

Offset: 1

Author

Ilya Gutkovskiy, May 04 2019

Keywords

• sign

Crossrefs

Cf. A001063.

Programs

Mathematica

terms = 21; A[] = 0; Do[A[x] = Normal[Integrate[1 + A[x/(1 + x) + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; Rest[CoefficientList[A[x], x] Range[0, terms]!]
a[n_] := a[n] = Sum[(-1)^(n - k - 1) Binomial[n - 2, k - 1] a[k] (n - 1)!/k!, {k, 1, n - 1}]; a[1] = 1; Table[a[n], {n, 1, 21}]

Formula

Recurrence: a(n+1) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*a(k)*n!/k!.