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A172349 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=4.

%I A172349 #4 Jul 05 2012 11:38:37
%S A172349 1,1,1,1,1,1,1,5,5,1,1,9,45,9,1,1,29,261,261,29,1,1,65,1885,3393,1885,
%T A172349 65,1,1,181,11765,68237,68237,11765,181,1,1,441,79821,1037673,3343613,
%U A172349 1037673,79821,441,1,1,1165,513765,18598293,134321005,134321005
%N A172349 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=4.
%C A172349 Start from the generalized Fibonacci sequence A006131 and its partial products c(n) = 1, 1, 1, 5, 45, 1305, 84825, 15353325, 6770816325, 7888001018625... Then t(n,k) = c(n)/(c(k)*c(n-k)).
%C A172349 Row sums are 1, 2, 3, 12, 65, 582, 7295, 160368, 5579485, 306868458, 26280601275,...
%e A172349 1;
%e A172349 1, 1;
%e A172349 1, 1, 1;
%e A172349 1, 5, 5, 1;
%e A172349 1, 9, 45, 9, 1;
%e A172349 1, 29, 261, 261, 29, 1;
%e A172349 1, 65, 1885, 3393, 1885, 65, 1;
%e A172349 1, 181, 11765, 68237, 68237, 11765, 181, 1;
%e A172349 1, 441, 79821, 1037673, 3343613, 1037673, 79821, 441, 1;
%e A172349 1, 1165, 513765, 18598293, 134321005, 134321005, 18598293, 513765, 1165, 1;
%e A172349 1, 2929, 3412285, 300963537, 6052711133, 13566421505, 6052711133, 300963537, 3412285, 2929, 1;
%t A172349 Clear[f, c, a, t];
%t A172349 f[0, a_] := 0; f[1, a_] := 1;
%t A172349 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172349 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172349 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172349 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172349 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172349 Cf. A010048 (m=1), A015109 (m=2), A172347 (m=3), A172350 (m=5).
%K A172349 nonn,tabl
%O A172349 0,8
%A A172349 _Roger L. Bagula_ and _Gary W. Adamson_, Feb 01 2010