A368708 -id:A368708 - cp's OEIS frontend

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

Joerg Arndt, Jan 04 2024

sign

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

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 *)

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

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

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

A368733 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -3).

Original entry on oeis.org

1, 1, 4, 22, 148, 1132, 9484, 85066, 804556, 7939738, 81128800, 853424464, 9201391456, 101327618056, 1136518296892, 12954283592578, 149770265417692, 1753615603901818, 20766700361401336, 248449277456597908, 3000039734827403608, 36532024054221028576, 448294209318801516064

Offset: 0

Vaclav Kotesovec, Jan 04 2024

nonn

Cf. A001181, A368708.

Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -3], {n, 0, 30}]
a[0] := 1; a[n_] := 3^n*Sum[(1/3)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1]/(Binomial[n + 1, 1]*Binomial[n + 1, 2]), {k, 1, n}]; Table[a[n], {n, 0, 22}] (* Detlef Meya, May 28 2024 *)

from sympy import hyperexpand
from sympy.functions import hyper
def A368733(n): return hyperexpand(hyper((-1-n,-n,1-n),(2,3),-3)) # Chai Wah Wu, Jan 04 2024

a(n) ~ (4 + 3^(4/3) + 3^(5/3))^(n + 5/3) / (3^(11/6) * Pi * n^4).

a(0) = 1, a(n) = 3^n*Sum_{k=1..n} (1/3)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1)/(binomial(n + 1, 1)*binomial(n + 1, 2)). - Detlef Meya, May 28 2024

cp's OEIS Frontend

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

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