A368709 - OEIS

%I #21 Jun 20 2024 05:29:29

%S 1,1,-1,-3,13,17,-241,121,5081,-13327,-106705,609589,1850661,

%T -23392159,-6796193,811545073,-1688514383,-25224774367,123764707231,

%U 650087614573,-6385330335427,-9591188592399,279171512779759,-318526766092183,-10665705513959287,40625771132796817

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

%F 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

%F 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

%t Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, 2], {n, 0, 30}] (* _Vaclav Kotesovec_, Jan 04 2024 *)

%t 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 *)

%o (SageMath)

%o def A368709(n): return PolyA359363(n, -2) // (-2) if n > 0 else 1

%o print([A368709(n) for n in range(0, 26)])  # _Peter Luschny_, Jan 04 2024

%o (Python)

%o def A368709(n):

%o     if n == 0: return 1

%o     return sum((-2)**k * v for k, v in enumerate(A359363Row(n))) // (-2)

%o print([A368709(n) for n in range(26)]) # _Peter Luschny_, Jan 04 2024

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

%K sign

%O 0,4

%A _Joerg Arndt_, Jan 04 2024