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A329076
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Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^n.
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3
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1, 0, 16, 72, 7008, 162000, 17555520, 1093527120, 140846184640, 16016249944800, 2550757928818680, 419682645514181280, 82389928294166805312, 17418502084657134228768, 4123280170924828458697152, 1054943518137131171386437600, 293933660095874311773617934720, 87968971083026619734709639853632
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OFFSET
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0,3
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COMMENTS
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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| = 2*n).
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LINKS
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FORMULA
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Conjecture: a(n) ~ 3 * 2^(3*n - 2) * n^(n-3) / Pi. - Vaclav Kotesovec, Nov 05 2019
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PROG
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(PARI) {a(n) = polcoef(polcoef((sum(k=-n, n, x^k)*sum(k=-n, n, y^k)-sum(k=-n+1, n-1, x^k)*sum(k=-n+1, n-1, y^k))^n, 0), 0)}
(PARI) {a(n) = polcoef(polcoef((sum(k=0, 2*n, (x^k+1/x^k)*(y^(2*n-k)+1/y^(2*n-k)))-x^(2*n)-1/x^(2*n)-y^(2*n)-1/y^(2*n))^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(n)^k*f(n-1)^(n-k), 0)^2)
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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