A368709 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], +2).

1, 1, -1, -3, 13, 17, -241, 121, 5081, -13327, -106705, 609589, 1850661, -23392159, -6796193, 811545073, -1688514383, -25224774367, 123764707231, 650087614573, -6385330335427, -9591188592399, 279171512779759, -318526766092183, -10665705513959287, 40625771132796817

Offset: 0

Author

Joerg Arndt, Jan 04 2024

Keywords

• sign

Crossrefs

Cf. A368708, A001181, A007724, A217800, A359363.

Programs

Mathematica

Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, 2], {n, 0, 30}] (* Vaclav Kotesovec, Jan 04 2024 *)
a[0] := 1; a[n_] := (-1)^n*2^(n + 1)/(n*(n + 1)^2)*Sum[(-1/2)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1], {k , 1, n}]; Table[a[n], {n, 0, 25}] (* Detlef Meya, May 29 2024 *)

Python

def A368709(n):
    if n == 0: return 1
    return sum((-2)**k * v for k, v in enumerate(A359363Row(n))) // (-2)
print([A368709(n) for n in range(26)]) # Peter Luschny, Jan 04 2024

SageMath

def A368709(n): return PolyA359363(n, -2) // (-2) if n > 0 else 1
print([A368709(n) for n in range(0, 26)])  # Peter Luschny, Jan 04 2024

Formula

a(n) = (-1/2)*B(n, -2) where B(n, x) are the Baxter polynomials with coefficients A359363, for n > 0. - Peter Luschny, Jan 04 2024

a(0) = 1, a(n) = (-1)^n*2^(n + 1)/(n*(n + 1)^2)*Sum_{k=1..n} (-1/2)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1). - Detlef Meya, May 29 2024