A346053 G.f. A(x) satisfies: A(x) = 1 - x * A(x/(1 - x)) / (1 - x)^3.

1, -1, -2, 0, 10, 25, -11, -301, -1040, -60, 17770, 95359, 146701, -1513837, -14210258, -53101500, 91834402, 2739189073, 19172894377, 46384729811, -498471972128, -7229201676480, -45007184571062, -40076612769641, 2435999270437801, 30321258115161275, 180120147363157438

Offset: 0

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..592

Crossrefs

Cf. A000587, A014619, A045501.

Programs

Mathematica

nmax = 26; A[] = 0; Do[A[x] = 1 - x A[x/(1 - x)]/(1 - x)^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n + 1, k + 2] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]

SageMath

@CachedFunction
def a(n): # a = A346053
    if (n==0): return 1
    else: return (-1)*sum(binomial(n+1, k+2)*a(k) for k in range(n))
[a(n) for n in range(51)] # G. C. Greubel, Dec 01 2022

Formula

a(n+1) = -Sum_{k=0..n} binomial(n+2,k+2) * a(k).