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

%I #10 Feb 21 2023 17:19:22

%S 0,1,4,13,40,117,324,853,2152,5245,12436,28845,65736,147685,327940,

%T 721189,1573192,3408237,7340436,15729085,33554920,71303701,150995524,

%U 318767733,671089320,1409286877,2952790804,6174016333,12884902792,26843546565,55834575876

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

%C The coefficients of the polygonal polynomials are 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 + 3*x - 1) / ((1 - 2*x)^2*(x - 1)^3).

%F a(n) = 8 + 4*n + n^2 + (n-4) * 2^(n+1). - _Vaclav Kotesovec_, Feb 21 2023

%p gf := (x*(-4*x^2 + 3*x - 1)) / ((1 - 2*x)^2*(x - 1)^3):

%p ser := series(gf, x, 32): seq(coeff(ser, x, n), n = 0..30);

%Y Cf. A139600, A360605.

%K nonn

%O 0,3

%A _Peter Luschny_, Feb 13 2023