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 A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2*(k+4)*x+((k-4)*x)^2) * (1+(k-4)*x+sqrt(1-2*(k+4)*x+((k-4)*x)^2)) )). 5
 1, 1, 6, 1, 7, 30, 1, 8, 51, 140, 1, 9, 74, 393, 630, 1, 10, 99, 736, 3139, 2772, 1, 11, 126, 1175, 7606, 25653, 12012, 1, 12, 155, 1716, 14499, 80464, 212941, 51480, 1, 13, 186, 2365, 24310, 183195, 864772, 1787607, 218790, 1, 14, 219, 3128, 37555, 352716, 2351805, 9400192, 15134931, 923780 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Antidiagonals n = 0..139, flattened FORMULA T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(2*j,j) * binomial(2*n+1,2*j). T(0,k) = 1, T(1,k) = k+6 and n * (2*n+1) * (4*n-3) * T(n,k) = (4*n-1) * (4*(k+4)*n^2-2*(k+4)*n-k-2) * T(n-1,k) - (k-4)^2 * (n-1) * (2*n-1) * (4*n+1) * T(n-2,k) for n > 1. - Seiichi Manyama, Aug 29 2020 For fixed k > 0, T(n,k) ~ (2 + sqrt(k))^(2*n + 3/2) / sqrt(8*k*Pi*n). - Vaclav Kotesovec, Aug 31 2020 EXAMPLE Square array begins: 1, 1, 1, 1, 1, 1, ... 6, 7, 8, 9, 10, 11, ... 30, 51, 74, 99, 126, 155, ... 140, 393, 736, 1175, 1716, 2365, ... 630, 3139, 7606, 14499, 24310, 37555, ... 2772, 25653, 80464, 183195, 352716, 610897, ... MATHEMATICA T[n_, k_] := Sum[If[k == 0, Boole[n == j], k^(n - j)] * Binomial[2*j, j] * Binomial[2*n + 1, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 25 2020 *) PROG (PARI) {T(n, k) = sum(j=0, n, k^(n-j)*binomial(2*j, j)*binomial(2*n+1, 2*j))} CROSSREFS Columns k=0..5 give A002457, A273055, A337370, A245927, A002458, A243947. Main diagonal gives A337387. Cf. A337389, A337464. Sequence in context: A046902 A204205 A143019 * A156921 A094214 A001622 Adjacent sequences: A337366 A337367 A337368 * A337370 A337371 A337372 KEYWORD nonn,tabl AUTHOR Seiichi Manyama, Aug 25 2020 STATUS approved

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Last modified August 10 01:33 EDT 2024. Contains 375044 sequences. (Running on oeis4.)