A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)*(y^3 + y + 1/y + 1/y^3) - (x + 1/x)*(y + 1/y))^(2*n).

1, 12, 588, 49440, 5187980, 597027312, 71962945824, 8923789535232, 1128795397492620, 144940851928720848, 18832163401980525168, 2470451402766989534256, 326667449725835512275488, 43485599433527022301377600, 5821983056232777427055717760

Offset: 0

Author

Seiichi Manyama, Nov 02 2019

Keywords

• nonn

Comments

Also number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 3).

*

|

*-- --*

| | |

*-- -- -- --*

| | | | |

*-- -- --P-- -- --*

| | | | |

*-- -- -- --*

| | |

*-- --*

|

*

Point P move to any position of * in the next step.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..400 (terms 0..185 from Vaclav Kotesovec)
• Vaclav Kotesovec, Recurrence of order 4 (conjectured)

Crossrefs

Row n=1 of A329066.

Cf. A002894, A094061, A254129.

Programs

PARI

{a(n) = polcoef(polcoef(((x^3+x+1/x+1/x^3)*(y^3+y+1/y+1/y^3)-(x+1/x)*(y+1/y))^(2*n), 0), 0)}

PARI

{a(n) = polcoef(polcoef((sum(k=0, 3, (x^k+1/x^k)*(y^(3-k)+1/y^(3-k)))-x^3-1/x^3-y^3-1/y^3)^(2*n), 0), 0)}

PARI

f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
a(n) = sum(k=0, 2*n, (-1)^k*binomial(2*n, k)*polcoef(f(1)^k*f(0)^(2*n-k), 0)^2)

Formula

Conjecture: a(n) ~ 3 * 144^n / (19*Pi*n). - Vaclav Kotesovec, Nov 04 2019