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A172358 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.

%I A172358 #15 May 10 2025 08:59:32
%S A172358 1,1,1,1,1,1,1,1,1,1,1,3,3,3,1,1,3,9,9,3,1,1,5,15,45,15,5,1,1,9,45,
%T A172358 135,135,45,9,1,1,11,99,495,495,495,99,11,1,1,19,209,1881,3135,3135,
%U A172358 1881,209,19,1,1,29,551,6061,18183,30305,18183,6061,551,29,1
%N A172358 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.
%C A172358 Start from the sequence A159284 and its partial products c(n) = 1, 1, 1, 1, 3, 9, 45, 405, 4455, 84645, 2454705, ... . Then T(n,k) = round( c(n)/(c(k)*c(n-k)) ).
%H A172358 G. C. Greubel, <a href="/A172358/b172358.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A172358 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) = f(n-2, q) + q*f(n-3, q), f(0,q) = 0, f(1,q) = f(2,q) = 1, and q = 2. - _G. C. Greubel_, May 09 2021
%e A172358 Triangle begins as:
%e A172358   1;
%e A172358   1,  1;
%e A172358   1,  1,   1;
%e A172358   1,  1,   1,    1;
%e A172358   1,  3,   3,    3,     1;
%e A172358   1,  3,   9,    9,     3,     1;
%e A172358   1,  5,  15,   45,    15,     5,     1;
%e A172358   1,  9,  45,  135,   135,    45,     9,    1;
%e A172358   1, 11,  99,  495,   495,   495,    99,   11,   1;
%e A172358   1, 19, 209, 1881,  3135,  3135,  1881,  209,  19,  1;
%e A172358   1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1;
%t A172358 f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, q]];
%t A172358 c[n_, q_]:= Product[f[j, q], {j,n}];
%t A172358 T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
%t A172358 Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 09 2021 *)
%o A172358 (Sage)
%o A172358 @CachedFunction
%o A172358 def f(n,q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, q)
%o A172358 def c(n,q): return product( f(j,q) for j in (1..n) )
%o A172358 def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
%o A172358 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 09 2021
%Y A172358 Cf. A172353 (q=1), this sequence (q=2), A172359 (q=4), A172360 (q=5).
%K A172358 nonn,tabl,less
%O A172358 0,12
%A A172358 _Roger L. Bagula_, Feb 01 2010
%E A172358 Definition corrected to give integral terms by _G. C. Greubel_, May 09 2021