A329024 - OEIS

%I #32 Nov 05 2019 14:09:10

%S 1,12,588,49440,5187980,597027312,71962945824,8923789535232,

%T 1128795397492620,144940851928720848,18832163401980525168,

%U 2470451402766989534256,326667449725835512275488,43485599433527022301377600,5821983056232777427055717760

%N 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).

%C 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).

%C          *

%C          |

%C       *-- --*

%C       |  |  |

%C    *-- -- -- --*

%C    |  |  |  |  |

%C *-- -- --P-- -- --*

%C    |  |  |  |  |

%C    *-- -- -- --*

%C       |  |  |

%C       *-- --*

%C          |

%C          *

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

%H Seiichi Manyama, <a href="/A329024/b329024.txt">Table of n, a(n) for n = 0..400</a> (terms 0..185 from Vaclav Kotesovec)

%H Vaclav Kotesovec, <a href="/A329024/a329024.txt">Recurrence of order 4 (conjectured)</a>

%F Conjecture: a(n) ~ 3 * 144^n / (19*Pi*n). - _Vaclav Kotesovec_, Nov 04 2019

%o (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)}

%o (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)}

%o (PARI) f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);

%o a(n) = sum(k=0, 2*n, (-1)^k*binomial(2*n, k)*polcoef(f(1)^k*f(0)^(2*n-k), 0)^2)

%Y Row n=1 of A329066.

%Y Cf. A002894, A094061, A254129.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 02 2019