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%I #24 Sep 08 2022 08:45:50
%S 1,1,1,1,3,1,1,15,15,1,1,55,275,55,1,1,231,4235,4235,231,1,1,903,
%T 69531,254947,69531,903,1,1,3655,1100155,16942387,16942387,1100155,
%U 3655,1,1,14535,17708475,1066050195,4477410819,1066050195,17708475,14535,1
%N A double product sequence based on a=2; f(n,a) = f(n-1,a) + a*f(n-2,a).
%H G. C. Greubel, <a href="/A173917/b173917.txt">Rows n = 0..50 of triangle, flattened</a>
%F c(n,a) = 1 if n = 0, Product_{i=1..n} f(i, a)*f(i+1, a) otherwise.
%F T(n,k) = Product_{i=1..k} ((q^(n+1-i)-1) / (q^i-1)) * ((q^(n+2-i)-1) / (q^(i+1)-1)) for 0 <= k <= n with q = -2 and the empty product 1 (k=0). - _Werner Schulte_, Nov 14 2018
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 15, 15, 1;
%e 1, 55, 275, 55, 1;
%e 1, 231, 4235, 4235, 231, 1;
%e 1, 903, 69531, 254947, 69531, 903, 1;
%e 1, 3655, 1100155, 16942387, 16942387, 1100155, 3655, 1;
%e 1, 14535, 17708475, 1066050195, 4477410819, 1066050195, 17708475, 14535, 1;
%e ...
%t f[0, a_] := 0; f[1, a_] := 1;
%t f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t c[n_, a_] := If[n == 0, 1, Product[f[i, a]*f[i + 1, a], {i, 1, n}]];
%t w[n_, m_, q_] := c[n, q]/(c[m, q]*c[n - m, q]);
%t Table[Table[Table[w[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 12}];
%t Table[Flatten[Table[Table[w[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 12}]
%t Table[Product[QBinomial[n+k, k+j, -2]/QBinomial[n+k-j, k, -2], {k, 0, 1}], {n, 0, 10}, {j, 0, n}]//Flatten (* _G. C. Greubel_, Nov 21 2018 *)
%o (PARI) T(n,k)={prod(i=0, k-1, (((-2)^(n-i)-1) / ((-2)^(i+1)-1) * ((-2)^(n+1-i)-1) / ((-2)^(i+2)-1)))} \\ _Andrew Howroyd_, Nov 12 2018
%o (Magma) q:=-2; [[k le 0 select 1 else (&*[((q^(n+1-i)-1)/(q^i-1))*((q^(n+2-i)-1)/(q^(i+1)-1)): i in [1..k]]) : k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 21 2018
%o (Sage) [[prod(q_binomial(n+k,k+j,-2)/q_binomial(n+k-j,k,-2) for k in (0..1)) for j in range(n+1)] for n in range(10)] # _G. C. Greubel_, Nov 21 2018
%Y Cf. A156916 (q=2).
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Mar 02 2010
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