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

%I #7 Feb 21 2023 17:19:13

%S 0,1,0,1,0,-3,8,-31,72,-195,448,-1071,2416,-5475,12120,-26719,58232,

%T -126243,271824,-582575,1242720,-2640899,5592360,-11806239,24855080,

%U -52195843,109362528,-228667311,477218512,-994205475,2067947128,-4294967391,8908080216

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

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

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

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

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

%p a := n -> (1/27)*(4-n)*(3*n + 2 + (-2)^(n + 1)):

%p seq(a(n), n = 0..32);

%Y Cf. A139600, A360606.

%K sign

%O 0,6

%A _Peter Luschny_, Feb 21 2023