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A172356 Triangle T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3), read by rows.

%I A172356 #8 May 10 2021 03:52:55
%S A172356 1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,3,6,6,3,1,1,4,12,24,12,4,1,1,6,24,72,
%T A172356 72,24,6,1,1,9,54,216,324,216,54,9,1,1,13,117,702,1404,1404,702,117,
%U A172356 13,1,1,19,247,2223,6669,8892,6669,2223,247,19,1
%N A172356 Triangle T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3), read by rows.
%H A172356 G. C. Greubel, <a href="/A172356/b172356.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A172356 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-3,q), f(0,q) = 0, f(1,q) = f(2,q) = 1, and q = 1.
%F A172356 T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3). - _G. C. Greubel_, May 09 2021
%e A172356 Triangle begins as:
%e A172356   1;
%e A172356   1,  1;
%e A172356   1,  1,   1;
%e A172356   1,  1,   1,    1;
%e A172356   1,  2,   2,    2,    1;
%e A172356   1,  3,   6,    6,    3,    1;
%e A172356   1,  4,  12,   24,   12,    4,    1;
%e A172356   1,  6,  24,   72,   72,   24,    6,    1;
%e A172356   1,  9,  54,  216,  324,  216,   54,    9,   1;
%e A172356   1, 13, 117,  702, 1404, 1404,  702,  117,  13,  1;
%e A172356   1, 19, 247, 2223, 6669, 8892, 6669, 2223, 247, 19, 1;
%t A172356 f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], q*f[n-1, q] + f[n-3, q]];
%t A172356 c[n_, q_]:= Product[f[j, q], {j,n}];
%t A172356 T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
%t A172356 Table[T[n, k, 1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 09 2021 *)
%o A172356 (Sage)
%o A172356 @CachedFunction
%o A172356 def f(n,q): return fibonacci(n) if (n<3) else q*f(n-1, q) + f(n-3, q)
%o A172356 def c(n,q): return product( f(j,q) for j in (1..n) )
%o A172356 def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
%o A172356 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 09 2021
%Y A172356 cf. A078012.
%K A172356 nonn,tabl,less
%O A172356 0,12
%A A172356 _Roger L. Bagula_, Feb 01 2010
%E A172356 Definition corrected to give integral terms by _G. C. Greubel_, May 09 2021