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A329078
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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the number of k-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = n).
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2
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1, 1, 1, 1, 0, 1, 1, 4, 0, 1, 1, 0, 8, 0, 1, 1, 36, 24, 12, 0, 1, 1, 0, 216, 0, 16, 0, 1, 1, 400, 1200, 588, 48, 20, 0, 1, 1, 0, 8840, 0, 1200, 0, 24, 0, 1, 1, 4900, 58800, 49440, 10200, 2100, 72, 28, 0, 1, 1, 0, 423640, 0, 165760, 0, 3336, 0, 32, 0, 1
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OFFSET
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0,8
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COMMENTS
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T(n,k) is the constant term in the expansion of (Sum_{j=0..n} (x^j + 1/x^j)*(y^(n-j) + 1/y^(n-j)) - x^n - 1/x^n - y^n - 1/y^n)^k for n > 0.
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LINKS
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 4, 0, 36, 0, ...
1, 0, 8, 24, 216, 1200, ...
1, 0, 12, 0, 588, 0, ...
1, 0, 16, 48, 1200, 10200, ...
1, 0, 20, 0, 2100, 0, ...
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PROG
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(PARI) {T(n, k) = if(n==0, 1, polcoef(polcoef((sum(j=0, n, (x^j+1/x^j)*(y^(n-j)+1/y^(n-j)))-x^n-1/x^n-y^n-1/y^n)^k, 0), 0))}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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