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 A321960 Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, .... 2
 1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 92, 57, 20, 5, 1, 0, 203, 426, 303, 116, 30, 6, 1, 0, 877, 2146, 1752, 744, 205, 42, 7, 1, 0, 4140, 11624, 10845, 5140, 1535, 330, 56, 8, 1, 0, 21147, 67146, 71139, 37676, 12300, 2820, 497, 72, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS For definitions and comments see A321964. LINKS FORMULA T(n, k) = A(n)[k] where A(n) is the Jacobi square of the sequence s(j) = n + j, j >= 0. EXAMPLE First few rows of the array start: [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007 [1] 1, 1, 2, 5, 15, 52, 203, 877, 4140, ... A000110 [2] 1, 2, 6, 22, 92, 426, 2146, 11624, 67146, ... A074664 [3] 1, 3, 12, 57, 303, 1752, 10845, 71139, 491064, ... A321959 [4] 1, 4, 20, 116, 744, 5140, 37676, 290224, 2334300, ... [5] 1, 5, 30, 205, 1535, 12300, 103975, 918785, 8434740, ... [6] 1, 6, 42, 330, 2820, 25662, 245358, 2443272, 25188870, ... [7] 1, 7, 56, 497, 4767, 48496, 516761, 5719399, 65369136, ... Seen as triangle: [0] 1; [1] 0, 1; [2] 0, 1, 1; [3] 0, 2, 2, 1; [4] 0, 5, 6, 3, 1; [5] 0, 15, 22, 12, 4, 1; [6] 0, 52, 92, 57, 20, 5, 1; [7] 0, 203, 426, 303, 116, 30, 6, 1; [8] 0, 877, 2146, 1752, 744, 205, 42, 7, 1; MAPLE # The function JacobiSquare is defined in A321964. s := n -> [seq(n+k, k = 0..9)]: Trow := n -> JacobiSquare(s(n)): for n from 0 to 7 do lprint(Trow(n)) od; MATHEMATICA nmax = 10; JacobiCF[a_, b_, p_:2] := Module[{m, k}, m = 1; For[k = Length[a], k >= 1, k--, m = 1 - b[[k]]*x - a[[k]]*x^p/m]; 1/m]; JacobiSquare[a_, p_: 2] := Module[{cf, ser}, cf = JacobiCF[a, a, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ser, x]]; s[n_] := Table[n + k, {k, 0, nmax}]; row[n_] := row[n] = JacobiSquare[s[n]]; T[_, 0] = 1; T[0, _] = 0; T[n_, k_] := row[n][[k + 1]]; Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 13 2019, after Peter Luschny in A321964 *) PROG (Sage) def JacobiCF(a, b, dim, p=2): m = 1 for k in range(dim-1, -1, -1): m = 1 - b(k)*x - a(k)*x^p/m return 1/m def JacobiGF(a, b, dim, p=2): cf = JacobiCF(a, b, dim, p) return cf.series(x, dim).list() def JacobiSquare(a, dim, p=2): cf = JacobiCF(a, a, dim, p) return cf.series(x, dim).list() def StieltjesGF(a, dim, p=2): return JacobiGF(a, lambda n: 0, dim, p) def Trow(n): return JacobiSquare(lambda k: n+k, 10) for n in (0..4): print(Trow(n)) CROSSREFS Rows of array: A000007, A000110, A074664, A321959. Columns include: A002378, A033445. Row sums of triangle: A321958. Cf. A321964. Sequence in context: A120059 A067347 A120568 * A189233 A242153 A065066 Adjacent sequences: A321957 A321958 A321959 * A321961 A321962 A321963 KEYWORD nonn,tabl AUTHOR Peter Luschny, Dec 27 2018 STATUS approved

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Last modified January 28 04:25 EST 2023. Contains 359850 sequences. (Running on oeis4.)