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 A358501 Irregular triangle read by rows. Coefficients of the polynomials (-1)^n*binomial(-x - 1, -x - n - 1) * binomial(n + x, x) * (n!)^2 in ascending order of powers. 0
 1, 1, 2, 1, 4, 12, 13, 6, 1, 36, 132, 193, 144, 58, 12, 1, 576, 2400, 4180, 3980, 2273, 800, 170, 20, 1, 14400, 65760, 129076, 143700, 100805, 46710, 14523, 3000, 395, 30, 1, 518400, 2540160, 5450256, 6787872, 5482456, 3034920, 1184153, 328986, 64743, 8820, 791, 42, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..48. FORMULA p(n, x) = (-1)^n*(-x - 1)!*(n + x)!/((-x - n - 1)!*x!). p(n, x) = (-1)^n*Pochhammer(-x - n, n) * Pochhammer(1 + x, n). EXAMPLE Polynomials p(n, x) begin: [0] 1; [1] -(-x - 1)*(x + 1); [2] (-x - 2)*(-x - 1)*(x + 1)*(x + 2); [3] -(-x - 3)*(-x - 2)*(-x - 1)*(x + 1)*(x + 2)*(x + 3); [4] (-x - 4)*(-x - 3)*(-x - 2)*(-x - 1)*(x + 1)*(x + 2)*(x + 3)*(x + 4); . Triangle T(n, k) begins: [0] 1; [1] 1, 2, 1; [2] 4, 12, 13, 6, 1; [3] 36, 132, 193, 144, 58, 12, 1; [4] 576, 2400, 4180, 3980, 2273, 800, 170, 20, 1; [5] 14400, 65760, 129076, 143700, 100805, 46710, 14523, 3000, 395, 30, 1; MAPLE p := (n, x) -> (-1)^n*binomial(-x - 1, -x - n - 1)*binomial(n + x, x)*(n!)^2: for n from 0 to 7 do seq(coeff(expand(p(n, x)), x, k), k = 0..2*n) od; CROSSREFS Cf. A001044 (row sums), A000007 (alternating row sums). Sequence in context: A371062 A354116 A064297 * A323493 A052661 A225117 Adjacent sequences: A358498 A358499 A358500 * A358502 A358503 A358504 KEYWORD nonn,tabf AUTHOR Peter Luschny, Nov 26 2022 STATUS approved

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Last modified May 25 03:39 EDT 2024. Contains 372782 sequences. (Running on oeis4.)