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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).

%I A329024 #32 Nov 05 2019 14:09:10
%S A329024 1,12,588,49440,5187980,597027312,71962945824,8923789535232,
%T A329024 1128795397492620,144940851928720848,18832163401980525168,
%U A329024 2470451402766989534256,326667449725835512275488,43485599433527022301377600,5821983056232777427055717760
%N 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).
%C A329024 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 A329024          *
%C A329024          |
%C A329024       *-- --*
%C A329024       |  |  |
%C A329024    *-- -- -- --*
%C A329024    |  |  |  |  |
%C A329024 *-- -- --P-- -- --*
%C A329024    |  |  |  |  |
%C A329024    *-- -- -- --*
%C A329024       |  |  |
%C A329024       *-- --*
%C A329024          |
%C A329024          *
%C A329024 Point P move to any position of * in the next step.
%H A329024 Seiichi Manyama, <a href="/A329024/b329024.txt">Table of n, a(n) for n = 0..400</a> (terms 0..185 from Vaclav Kotesovec)
%H A329024 Vaclav Kotesovec, <a href="/A329024/a329024.txt">Recurrence of order 4 (conjectured)</a>
%F A329024 Conjecture: a(n) ~ 3 * 144^n / (19*Pi*n). - _Vaclav Kotesovec_, Nov 04 2019
%o A329024 (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 A329024 (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 A329024 (PARI) f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
%o A329024 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 A329024 Row n=1 of A329066.
%Y A329024 Cf. A002894, A094061, A254129.
%K A329024 nonn
%O A329024 0,2
%A A329024 _Seiichi Manyama_, Nov 02 2019