A360606 The polygonal polynomials evaluated at x = 1/2 and normalized with 2^n.

0, 1, 4, 13, 40, 117, 324, 853, 2152, 5245, 12436, 28845, 65736, 147685, 327940, 721189, 1573192, 3408237, 7340436, 15729085, 33554920, 71303701, 150995524, 318767733, 671089320, 1409286877, 2952790804, 6174016333, 12884902792, 26843546565, 55834575876

Offset

0,3

Comments

The coefficients of the polygonal polynomials are antidiagonals of A139600.

Links

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

Index entries for linear recurrences with constant coefficients, signature (7,-19,25,-16,4).

Formula

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

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

a(n) = 8 + 4*n + n^2 + (n-4) * 2^(n+1). - Vaclav Kotesovec, Feb 21 2023

a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5) for n > 4. - Chai Wah Wu, Apr 16 2025

Maple

gf := (x*(-4*x^2 + 3*x - 1)) / ((1 - 2*x)^2*(x - 1)^3):
ser := series(gf, x, 32): seq(coeff(ser, x, n), n = 0..30);

Mathematica

A360606[n_] := n^2 + 4*n + 2^(n+1)*(n-4) + 8; Array[A360606, 35, 0] (* Paolo Xausa, Jun 25 2026 *)
(* Alternative: *)
LinearRecurrence[{7, -19, 25, -16, 4}, {0, 1, 4, 13, 40}, 35] (* Paolo Xausa, Jun 25 2026 *)

Crossrefs

Cf. A139600, A360605.

Sequence in context: A272581 A342159 A191132 * A119915 A307577 A137744

Adjacent sequences: A360603 A360604 A360605 * A360607 A360608 A360609

Keyword

nonn,easy,changed

Author

Peter Luschny, Feb 13 2023

Status

approved