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 A365673 Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences. 6
 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 15, 8, 1, 1, 1, 5, 34, 105, 16, 1, 1, 1, 6, 61, 496, 945, 32, 1, 1, 1, 7, 96, 1385, 11056, 10395, 64, 1, 1, 1, 8, 139, 2976, 50521, 349504, 135135, 128, 1, 1, 1, 9, 190, 5473, 151416, 2702765, 14873104, 2027025, 256, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Using polygonal numbers as weights, a recursion for triangles is defined, whose main diagonals represents a family of sequences, which include, among others, the powers of 2, the double factorial of odd numbers, the reduced tangent numbers, and the Euler numbers. Apart from the edge cases k = 0 and k = n the recursion is T(n, k) = w(n, k) * T(n, k - 1) + T(n - 1, k). T(n, 0) = 1 and T(n, n) = T(n, n-1) if n > 0. The weights w(n, k) identical to 1 yield the recursion of the Catalan triangle A009766 (with main diagonal the Catalan numbers). Here the polygonal numbers are used as weights in the form w(n, k) = p(s, n - k + 1), where the parameter s is the number of sides of the polygon and p(s, n) = ((s-2) * n^2 - (s-4) * n) / 2, see A317302. LINKS Table of n, a(n) for n=0..65. Wikipedia, Polygonal number. EXAMPLE Array A(n, k) starts: (polygon|diagonal|triangle) [0] 1, 1, 1, 1, 1, 1, 1, ... A258837 A000012 [1] 1, 1, 2, 4, 8, 16, 32, ... A080956 A011782 [2] 1, 1, 3, 15, 105, 945, 10395, ... A001477 A001147 A001498 [3] 1, 1, 4, 34, 496, 11056, 349504, ... A000217 A002105 A365674 [4] 1, 1, 5, 61, 1385, 50521, 2702765, ... A000290 A000364 A060058 [5] 1, 1, 6, 96, 2976, 151416, 11449296, ... A000326 A126151 A366138 [6] 1, 1, 7, 139, 5473, 357721, 34988647, ... A000384 A126156 A365672 [7] 1, 1, 8, 190, 9080, 725320, 87067520, ... A000566 A366150 A366149 [8] 1, 1, 9, 249, 14001, 1322001, 188106489, ... A000567 A054556 A366137 MAPLE poly := (s, n) -> ((s - 2) * n^2 - (s - 4) * n) / 2: T := proc(s, n, k) option remember; if k = 0 then 1 else if k = n then T(s, n, k-1) else poly(s, n - k + 1) * T(s, n, k - 1) + T(s, n - 1, k) fi fi end: for n from 0 to 8 do A := (n, k) -> T(n, k, k): seq(A(n, k), k = 0..9) od; # Alternative, using continued fractions: A := proc(p, L) local CF, poly, k, m, P, ser; poly := (s, n) -> ((s - 2)*n^2 - (s - 4)*n)/2; CF := 1 + x; for k from 1 to L do m := L - k + 1; P := poly(p, m); CF := 1/(1 - P*x*CF) od; ser := series(CF, x, L); seq(coeff(ser, x, m), m = 0..L-1) end: for p from 0 to 8 do lprint(A(p, 8)) od; MATHEMATICA poly[s_, n_] := ((s - 2) * n^2 - (s - 4) * n) / 2; T[s_, n_, k_] := T[s, n, k] = If[k == 0, 1, If[k == n, T[s, n, k - 1], poly[s, n - k + 1] * T[s, n, k - 1] + T[s, n - 1, k]]]; A[n_, k_] := T[n, k, k]; Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2023, from first Maple program *) PROG (Python) from functools import cache @cache def T(s, n, k): if k == 0: return 1 if k == n: return T(s, n, k - 1) p = (n - k + 1) * ((s - 2) * (n - k + 1) - (s - 4)) // 2 return p * T(s, n, k - 1) + T(s, n - 1, k) def A(n, k): return T(n, k, k) for n in range(9): print([A(n, k) for k in range(9)]) (PARI) A(p, n) = { my(CF = 1 + x, poly(s, n) = ((s - 2)*n^2 - (s - 4)*n)/2, m, P ); for(k = 1, n, m = n - k + 1; P = poly(p, m); CF = 1/(1 - P*x*CF) ); Vec(CF + O(x^(n))) } for(p = 0, 8, print(A(p, 8))) \\ Michel Marcus and Peter Luschny, Oct 02 2023 CROSSREFS Poly weights: A258837, A080956, A001477, A000217, A000290, A000326, A000384. Rows: A000012, A011782, A001147, A002105, A000364, A126151, A126156, A366150. Triangles: A001498, A365674, A060058, A366138, A365672, A366149. Cf. A009766, A366137 (central diagonal), A317302 (table of polygonal numbers). Cf. A112934, A303943, A305532, A305533. Sequence in context: A293991 A288638 A261494 * A349574 A168377 A122867 Adjacent sequences: A365670 A365671 A365672 * A365674 A365675 A365676 KEYWORD nonn,tabl AUTHOR Peter Luschny, Sep 30 2023 STATUS approved

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Last modified August 11 11:30 EDT 2024. Contains 375068 sequences. (Running on oeis4.)