%I #38 Nov 27 2023 06:22:40

%S 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 61,496,945,32,1,1,1,7,96,1385,11056,10395,64,1,1,1,8,139,2976,50521,

%U 349504,135135,128,1,1,1,9,190,5473,151416,2702765,14873104,2027025,256,1

%N Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences.

%C 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 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.

%e Array A(n, k) starts: (polygon|diagonal|triangle)

%e [0] 1, 1, 1, 1, 1, 1, 1, ... A258837 A000012

%e [1] 1, 1, 2, 4, 8, 16, 32, ... A080956 A011782

%e [2] 1, 1, 3, 15, 105, 945, 10395, ... A001477 A001147 A001498

%e [3] 1, 1, 4, 34, 496, 11056, 349504, ... A000217 A002105 A365674

%e [4] 1, 1, 5, 61, 1385, 50521, 2702765, ... A000290 A000364 A060058

%e [5] 1, 1, 6, 96, 2976, 151416, 11449296, ... A000326 A126151 A366138

%e [6] 1, 1, 7, 139, 5473, 357721, 34988647, ... A000384 A126156 A365672

%e [7] 1, 1, 8, 190, 9080, 725320, 87067520, ... A000566 A366150 A366149

%e [8] 1, 1, 9, 249, 14001, 1322001, 188106489, ... A000567

%e A054556 A366137

%p poly := (s, n) -> ((s - 2) * n^2 - (s - 4) * n) / 2:

%p 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 for n from 0 to 8 do A := (n, k) -> T(n, k, k): seq(A(n, k), k = 0..9) od;

%p # Alternative, using continued fractions:

%p A := proc(p, L) local CF, poly, k, m, P, ser;

%p poly := (s, n) -> ((s - 2)*n^2 - (s - 4)*n)/2;

%p CF := 1 + x;

%p for k from 1 to L do

%p m := L - k + 1;

%p P := poly(p, m);

%p CF := 1/(1 - P*x*CF)

%p od;

%p ser := series(CF, x, L);

%p seq(coeff(ser, x, m), m = 0..L-1)

%p end:

%p for p from 0 to 8 do lprint(A(p, 8)) od;

%t poly[s_, n_] := ((s - 2) * n^2 - (s - 4) * n) / 2;

%t 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 A[n_, k_] := T[n, k, k];

%t Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 27 2023, from first Maple program *)

%o (Python)

%o from functools import cache

%o @cache

%o def T(s, n, k):

%o if k == 0: return 1

%o if k == n: return T(s, n, k - 1)

%o p = (n - k + 1) * ((s - 2) * (n - k + 1) - (s - 4)) // 2

%o return p * T(s, n, k - 1) + T(s, n - 1, k)

%o def A(n, k): return T(n, k, k)

%o for n in range(9): print([A(n, k) for k in range(9)])

%o (PARI)

%o A(p, n) = {

%o my(CF = 1 + x,

%o poly(s, n) = ((s - 2)*n^2 - (s - 4)*n)/2,

%o m, P

%o );

%o for(k = 1, n,

%o m = n - k + 1;

%o P = poly(p, m);

%o CF = 1/(1 - P*x*CF)

%o );

%o Vec(CF + O(x^(n)))

%o }

%o for(p = 0, 8, print(A(p, 8)))

%o \\ _Michel Marcus_ and _Peter Luschny_, Oct 02 2023

%Y Poly weights: A258837, A080956, A001477, A000217, A000290, A000326, A000384.

%Y Rows: A000012, A011782, A001147, A002105, A000364, A126151, A126156, A366150.

%Y Triangles: A001498, A365674, A060058, A366138, A365672, A366149.

%Y Cf. A009766, A366137 (central diagonal), A317302 (table of polygonal numbers).

%Y Cf. A112934, A303943, A305532, A305533.

%K nonn,tabl

%O 0,9

%A _Peter Luschny_, Sep 30 2023