A396833 The constant coefficient of (x + y + y^-1 + x^-1 + x^-1*y^-1)^n.

1, 0, 4, 6, 36, 120, 490, 2100, 8260, 36960, 151704, 674520, 2884266, 12756744, 55922724, 247843596, 1101131460, 4907663904, 21972389296, 98527788216, 443467240216, 1999155993120, 9036349011144, 40913243644416, 185603468944426, 843338309520720, 3837872226246400

Offset

0,3

Comments

The Newton polytope of the Laurent polynomial x + y + y^-1 + x^-1 + x^-1*y^-1 is a reflexive polytope.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

From Vaclav Kotesovec, Jun 09 2026: (Start)

Recurrence: n^2*(115*n - 213)*a(n) = -2*(n-1)*(115*n^2 - 213*n + 119)*a(n-1) + (1955*n^3 - 7531*n^2 + 9184*n - 3612)*a(n-2) + (n-2)*(7015*n^2 - 20008*n + 12257)*a(n-3) + 43*(n-3)*(n-2)*(115*n - 98)*a(n-4).

a(n) ~ sqrt(37 - 52*(2/(23*(21551 + 2595*sqrt(69))))^(1/3) + 2*(2/23)^(2/3)*(21551 + 2595*sqrt(69))^(1/3)) * (((-1 + ((997 - 69*sqrt(69))/2)^(1/3) + ((997 + 69*sqrt(69))/2)^(1/3))/3)^n / (2*sqrt(15)*Pi*n)). (End)

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(2*k,k) * binomial(k,n-2*k). - Seiichi Manyama, Jun 10 2026

Mathematica

Table[SeriesCoefficient[Expand[(x + y + y^-1 + x^-1 + x^-1*y^-1)^n], {x, 0, 0}, {y, 0, 0}], {n, 0, 25}] (* Vaclav Kotesovec, Jun 09 2026 *)

Prog

(PARI) a(n) = polcoef(polcoef((x + y + y^-1 + x^-1 + x^-1*y^-1)^n, 0), 0); \\ Michel Marcus, Jun 10 2026

Crossrefs

Similar to A356521.

Sequence in context: A375412 A092187 A092765 * A056315 A103234 A074061

Adjacent sequences: A396823 A396828 A396829 * A396836 A396837 A396850

Keyword

nonn,new

Author

F. Chapoton, Jun 07 2026

Status

approved