A329077 Constant term in the expansion of ((Sum_{k=-3..3} x^k) * (Sum_{k=-3..3} y^k) - (Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k))^n.

1, 0, 24, 72, 3336, 34800, 912840, 15661520, 355423880, 7241240160, 160151370624, 3461028611040, 76789098028104, 1700195813892576, 38037857914721808, 853169553940415712, 19240825799184080520, 435267116844063531456, 9882232970998312871232

Offset

0,3

Comments

Also number of n-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 6).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500 (terms 0..230 from Vaclav Kotesovec)

Wikipedia, Taxicab geometry.

Formula

Conjecture: a(n) ~ 6 * 24^n / (73*Pi*n). - Vaclav Kotesovec, Nov 04 2019

Prog

(PARI) {a(n) = polcoef(polcoef((sum(k=-3, 3, x^k)*sum(k=-3, 3, y^k)-sum(k=-2, 2, x^k)*sum(k=-2, 2, y^k))^n, 0), 0)}
(PARI) {a(n) = polcoef(polcoef((sum(k=0, 6, (x^k+1/x^k)*(y^(6-k)+1/y^(6-k)))-x^6-1/x^6-y^6-1/y^6)^n, 0), 0)}
(PARI) f(n) = (x^(n+1)-1/x^n)/(x-1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(3)^k*f(2)^(n-k), 0)^2)

Crossrefs

Row n=3 of A329074.

Cf. A002894, A094061, A329024, A329067, A329075.

Sequence in context: A192833 A292353 A189540 * A339883 A042128 A043406

Adjacent sequences: A329074 A329075 A329076 * A329078 A329079 A329080

Keyword

nonn,walk

Author

Seiichi Manyama, Nov 04 2019

Status

approved