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A172339 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=3.

%I A172339 #5 Jul 05 2012 11:37:56
%S A172339 1,1,1,1,3,1,1,10,10,1,1,33,110,33,1,1,109,1199,1199,109,1,1,360,
%T A172339 13080,43164,13080,360,1,1,1189,142680,1555212,1555212,142680,1189,1,
%U A172339 1,3927,1556401,56030436,185070228,56030436,1556401,3927,1,1,12970,16977730
%N A172339 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=3.
%C A172339 Build with the recipe of A010048 (m=1) and A099927 (m=2).
%C A172339 Start from the generalized Fibonacci sequence A006190 and its partial products c(n) = 1, 1, 3, 30, 990, 107910, 38847600, 46189796400,... Then t(n,k) = c(n)/(c(k)*c(n-k)).
%C A172339 Row sums are 1, 2, 5, 22, 178, 2618, 70046, 3398164, 300251758, 48114604076,
%C A172339 14041125439724,...
%e A172339 1;
%e A172339 1, 1;
%e A172339 1, 3, 1;
%e A172339 1, 10, 10, 1;
%e A172339 1, 33, 110, 33, 1;
%e A172339 1, 109, 1199, 1199, 109, 1;
%e A172339 1, 360, 13080, 43164, 13080, 360, 1;
%e A172339 1, 1189, 142680, 1555212, 1555212, 142680, 1189, 1;
%e A172339 1, 3927, 1556401, 56030436, 185070228, 56030436, 1556401, 3927, 1;
%e A172339 1, 12970, 16977730, 2018652097, 22021659240, 22021659240, 2018652097, 16977730, 12970, 1;
%t A172339 Clear[f, c, a, t];
%t A172339 f[0, a_] := 0; f[1, a_] := 1;
%t A172339 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172339 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172339 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172339 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172339 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172339 Cf. A010048 (m=1), A099927 (m=2), A034802 (m=4), A172342 (m=5).
%K A172339 nonn,tabl,easy
%O A172339 0,5
%A A172339 _Roger L. Bagula_, Feb 01 2010