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A172360 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 A172360 #17 May 10 2025 08:59:42
%S A172360 1,1,1,1,1,1,1,1,1,1,1,6,6,6,1,1,6,36,36,6,1,1,11,66,396,66,11,1,1,36,
%T A172360 396,2376,2376,396,36,1,1,41,1476,16236,16236,16236,1476,41,1,1,91,
%U A172360 3731,134316,246246,246246,134316,3731,91,1,1,221,20111,824551,4947306,9070061,4947306,824551,20111,221,1
%N A172360 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 A172360 Start from the sequence 0, 1, 1, 1, 6, 6, 11, 36, 41, 91, 221, 296, 676, 1401, 2156, ..., f(n) = f(n-2) + 5*f(n-3), and its partial products c(n) = 1, 1, 1, 1, 6, 36, 396, 14256, 584496, 53189136, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).
%H A172360 G. C. Greubel, <a href="/A172360/b172360.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A172360 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 = 5. - _G. C. Greubel_, May 09 2021
%e A172360 Triangle begins as:
%e A172360   1;
%e A172360   1,   1;
%e A172360   1,   1,     1;
%e A172360   1,   1,     1,      1;
%e A172360   1,   6,     6,      6,       1;
%e A172360   1,   6,    36,     36,       6,       1;
%e A172360   1,  11,    66,    396,      66,      11,       1;
%e A172360   1,  36,   396,   2376,    2376,     396,      36,      1;
%e A172360   1,  41,  1476,  16236,   16236,   16236,    1476,     41,     1;
%e A172360   1,  91,  3731, 134316,  246246,  246246,  134316,   3731,    91,   1;
%e A172360   1, 221, 20111, 824551, 4947306, 9070061, 4947306, 824551, 20111, 221, 1;
%t A172360 f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, q]];
%t A172360 c[n_, q_]:= Product[f[j, q], {j,n}];
%t A172360 T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
%t A172360 Table[T[n, k, 5], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 09 2021 *)
%o A172360 (Sage)
%o A172360 @CachedFunction
%o A172360 def f(n,q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, q)
%o A172360 def c(n,q): return product( f(j,q) for j in (1..n) )
%o A172360 def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
%o A172360 flatten([[T(n,k,5) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 09 2021
%Y A172360 Cf. A172353 (q=1), A172358 (q=2), A172359 (q=4), this sequence (q=5).
%K A172360 nonn,tabl,less
%O A172360 0,12
%A A172360 _Roger L. Bagula_, Feb 01 2010
%E A172360 Definition corrected to give integral terms by _G. C. Greubel_, May 09 2021