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A172363 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of A003269.

%I A172363 #14 May 09 2021 09:50:05
%S A172363 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,1,1,3,6,6,6,3,1,1,4,12,24,24,
%T A172363 12,4,1,1,5,20,60,120,60,20,5,1,1,7,35,140,420,420,140,35,7,1,1,10,70,
%U A172363 350,1400,2100,1400,350,70,10,1
%N A172363 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of A003269.
%H A172363 G. C. Greubel, <a href="/A172363/b172363.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A172363 T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*f(n-1, q) + f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 1. - _G. C. Greubel_, May 08 2021
%e A172363 Triangle begins as:
%e A172363   1;
%e A172363   1,  1;
%e A172363   1,  1,  1;
%e A172363   1,  1,  1,   1;
%e A172363   1,  1,  1,   1,    1;
%e A172363   1,  2,  2,   2,    2,    1;
%e A172363   1,  3,  6,   6,    6,    3,    1;
%e A172363   1,  4, 12,  24,   24,   12,    4,   1;
%e A172363   1,  5, 20,  60,  120,   60,   20,   5,  1;
%e A172363   1,  7, 35, 140,  420,  420,  140,  35,  7,  1;
%e A172363   1, 10, 70, 350, 1400, 2100, 1400, 350, 70, 10, 1;
%t A172363 f[n_, q_]:= f[n, q]= If[n==0,0,If[n<4, 1, q*f[n-1, q] + f[n-4, q]]];
%t A172363 c[n_, q_]:= Product[f[j, q], {j,n}];
%t A172363 T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
%t A172363 Table[T[n, k, 1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 08 2021 *)
%o A172363 (Sage)
%o A172363 @CachedFunction
%o A172363 def f(n,q): return 0 if (n==0) else 1 if (n<4) else q*f(n-1, q) + f(n-4, q)
%o A172363 def c(n,q): return product( f(j,q) for j in (1..n) )
%o A172363 def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
%o A172363 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 08 2021
%Y A172363 Cf. A003269 (f(n,1)), A172363 (q=1), A172364 (q=3).
%K A172363 nonn,tabl
%O A172363 0,17
%A A172363 _Roger L. Bagula_, Feb 01 2010
%E A172363 Definition corrected to give integral terms, _G. C. Greubel_, May 08 2021