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 A329074 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k. 6
 1, 1, 1, 1, 0, 1, 1, 8, 0, 1, 1, 24, 16, 0, 1, 1, 216, 48, 24, 0, 1, 1, 1200, 1200, 72, 32, 0, 1, 1, 8840, 10200, 3336, 96, 40, 0, 1, 1, 58800, 165760, 34800, 7008, 120, 48, 0, 1, 1, 423640, 2032800, 912840, 82800, 12600, 144, 56, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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| = 2*n). T(n,k) is the constant term in the expansion of (Sum_{j=0..2*n} (x^j + 1/x^j)*(y^(2*n-j) + 1/y^(2*n-j)) - x^(2*n) - 1/x^(2*n) - y^(2*n) - 1/y^(2*n))^k for n > 0. LINKS Seiichi Manyama, Antidiagonals n = 0..100, flattened Wikipedia, Taxicab geometry. FORMULA T(0,k) = 1^k = 1. See the second code written in PARI. EXAMPLE Square array begins:    1, 1,  1,   1,     1,      1, ...    1, 0,  8,  24,   216,   1200, ...    1, 0, 16,  48,  1200,  10200, ...    1, 0, 24,  72,  3336,  34800, ...    1, 0, 32,  96,  7008,  82800, ...    1, 0, 40, 120, 12600, 162000, ... PROG (PARI) {T(n, k) = if(n==0, 1, polcoef(polcoef((sum(j=0, 2*n, (x^j+1/x^j)*(y^(2*n-j)+1/y^(2*n-j)))-x^(2*n)-1/x^(2*n)-y^(2*n)-1/y^(2*n))^k, 0), 0))} (PARI) f(n) = (x^(n+1)-1/x^n)/(x-1); T(n, k) = if(n==0, 1, sum(j=0, k, (-1)^(k-j)*binomial(k, j)*polcoef(f(n)^j*f(n-1)^(k-j), 0)^2)) CROSSREFS Rows n=0-3 give A000012, A094061, A329075, A329077. Main diagonal gives A329076. Cf. A329066. Sequence in context: A061847 A307224 A309595 * A296434 A164790 A250219 Adjacent sequences:  A329071 A329072 A329073 * A329075 A329076 A329077 KEYWORD nonn,tabl,walk AUTHOR Seiichi Manyama, Nov 03 2019 STATUS approved

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Last modified April 5 00:43 EDT 2020. Contains 333238 sequences. (Running on oeis4.)