This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365673 #38 Nov 27 2023 06:22:40
%S A365673 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,
%T A365673 61,496,945,32,1,1,1,7,96,1385,11056,10395,64,1,1,1,8,139,2976,50521,
%U A365673 349504,135135,128,1,1,1,9,190,5473,151416,2702765,14873104,2027025,256,1
%N A365673 Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences.
%C A365673 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.
%C A365673 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.
%C A365673 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.
%H A365673 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.
%e A365673 Array A(n, k) starts: (polygon|diagonal|triangle)
%e A365673 [0] 1, 1, 1, 1, 1, 1, 1, ... A258837 A000012
%e A365673 [1] 1, 1, 2, 4, 8, 16, 32, ... A080956 A011782
%e A365673 [2] 1, 1, 3, 15, 105, 945, 10395, ... A001477 A001147 A001498
%e A365673 [3] 1, 1, 4, 34, 496, 11056, 349504, ... A000217 A002105 A365674
%e A365673 [4] 1, 1, 5, 61, 1385, 50521, 2702765, ... A000290 A000364 A060058
%e A365673 [5] 1, 1, 6, 96, 2976, 151416, 11449296, ... A000326 A126151 A366138
%e A365673 [6] 1, 1, 7, 139, 5473, 357721, 34988647, ... A000384 A126156 A365672
%e A365673 [7] 1, 1, 8, 190, 9080, 725320, 87067520, ... A000566 A366150 A366149
%e A365673 [8] 1, 1, 9, 249, 14001, 1322001, 188106489, ... A000567
%e A365673 A054556 A366137
%p A365673 poly := (s, n) -> ((s - 2) * n^2 - (s - 4) * n) / 2:
%p A365673 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:
%p A365673 for n from 0 to 8 do A := (n, k) -> T(n, k, k): seq(A(n, k), k = 0..9) od;
%p A365673 # Alternative, using continued fractions:
%p A365673 A := proc(p, L) local CF, poly, k, m, P, ser;
%p A365673 poly := (s, n) -> ((s - 2)*n^2 - (s - 4)*n)/2;
%p A365673 CF := 1 + x;
%p A365673 for k from 1 to L do
%p A365673 m := L - k + 1;
%p A365673 P := poly(p, m);
%p A365673 CF := 1/(1 - P*x*CF)
%p A365673 od;
%p A365673 ser := series(CF, x, L);
%p A365673 seq(coeff(ser, x, m), m = 0..L-1)
%p A365673 end:
%p A365673 for p from 0 to 8 do lprint(A(p, 8)) od;
%t A365673 poly[s_, n_] := ((s - 2) * n^2 - (s - 4) * n) / 2;
%t A365673 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]]];
%t A365673 A[n_, k_] := T[n, k, k];
%t A365673 Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 27 2023, from first Maple program *)
%o A365673 (Python)
%o A365673 from functools import cache
%o A365673 @cache
%o A365673 def T(s, n, k):
%o A365673 if k == 0: return 1
%o A365673 if k == n: return T(s, n, k - 1)
%o A365673 p = (n - k + 1) * ((s - 2) * (n - k + 1) - (s - 4)) // 2
%o A365673 return p * T(s, n, k - 1) + T(s, n - 1, k)
%o A365673 def A(n, k): return T(n, k, k)
%o A365673 for n in range(9): print([A(n, k) for k in range(9)])
%o A365673 (PARI)
%o A365673 A(p, n) = {
%o A365673 my(CF = 1 + x,
%o A365673 poly(s, n) = ((s - 2)*n^2 - (s - 4)*n)/2,
%o A365673 m, P
%o A365673 );
%o A365673 for(k = 1, n,
%o A365673 m = n - k + 1;
%o A365673 P = poly(p, m);
%o A365673 CF = 1/(1 - P*x*CF)
%o A365673 );
%o A365673 Vec(CF + O(x^(n)))
%o A365673 }
%o A365673 for(p = 0, 8, print(A(p, 8)))
%o A365673 \\ _Michel Marcus_ and _Peter Luschny_, Oct 02 2023
%Y A365673 Poly weights: A258837, A080956, A001477, A000217, A000290, A000326, A000384.
%Y A365673 Rows: A000012, A011782, A001147, A002105, A000364, A126151, A126156, A366150.
%Y A365673 Triangles: A001498, A365674, A060058, A366138, A365672, A366149.
%Y A365673 Cf. A009766, A366137 (central diagonal), A317302 (table of polygonal numbers).
%Y A365673 Cf. A112934, A303943, A305532, A305533.
%K A365673 nonn,tabl
%O A365673 0,9
%A A365673 _Peter Luschny_, Sep 30 2023