A360605 The polygonal polynomials evaluated at x = -1/2 and normalized with (-2)^n.

0, 1, 0, 1, 0, -3, 8, -31, 72, -195, 448, -1071, 2416, -5475, 12120, -26719, 58232, -126243, 271824, -582575, 1242720, -2640899, 5592360, -11806239, 24855080, -52195843, 109362528, -228667311, 477218512, -994205475, 2067947128, -4294967391, 8908080216

Offset

0,6

Comments

The coefficients of the polygonal polynomials are the antidiagonals of A139600.

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-1,5,1,-8,4).

Formula

a(n) = (-2)^n * Sum_{k=0..n} A139600(n, k) * (-2)^(-k).

a(n) = [x^n] x*(4*x^2 - x - 1) / ((2*x + 1)^2*(x - 1)^3).

a(n) = (4 - n)*(3*n + 2 + (-2)^(n + 1)) / 27.

a(n) = - a(n-1) + 5*a(n-2) + a(n-3) - 8*a(n-4) + 4*a(n-5) for n > 4. - Chai Wah Wu, Apr 16 2025

Maple

a := n -> (1/27)*(4-n)*(3*n + 2 + (-2)^(n + 1)):
seq(a(n), n = 0..32);

Mathematica

A360605[n_] := (4-n)*(3*n + (-2)^(n+1) + 2)/27; Array[A360605, 35, 0] (* Paolo Xausa, Jun 25 2026 *)
(* Alternative: *)
LinearRecurrence[{-1, 5, 1, -8, 4}, {0, 1, 0, 1, 0}, 35] (* Paolo Xausa, Jun 25 2026 *)

Crossrefs

Cf. A139600, A360606.

Sequence in context: A379905 A066165 A323775 * A119838 A148889 A148890

Adjacent sequences: A360602 A360603 A360604 * A360606 A360607 A360608

Keyword

sign,easy,changed

Author

Peter Luschny, Feb 21 2023

Status

approved