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A172347 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 A172347 #4 Jul 05 2012 11:46:22
%S A172347 1,1,1,1,1,1,1,4,4,1,1,7,28,7,1,1,19,133,133,19,1,1,40,760,1330,760,
%T A172347 40,1,1,97,3880,18430,18430,3880,97,1,1,217,21049,210490,571330,
%U A172347 210490,21049,217,1,1,508,110236,2673223,15275560,15275560,2673223,110236,508
%N A172347 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 A172347 Start from the generalized Fibonacci sequence A006130 and its partial products c(n) = 1, 1, 1, 4, 28, 532, 21280, 2064160, 447922720, 227544741760... Then t(n,k) = c(n)/(c(k)*c(n-k)).
%C A172347 Row sums are 1, 2, 3, 10, 44, 306, 2932, 44816, 1034844, 36119056, 1882089488,...
%e A172347 1;
%e A172347 1, 1;
%e A172347 1, 1, 1;
%e A172347 1, 4, 4, 1;
%e A172347 1, 7, 28, 7, 1;
%e A172347 1, 19, 133, 133, 19, 1;
%e A172347 1, 40, 760, 1330, 760, 40, 1;
%e A172347 1, 97, 3880, 18430, 18430, 3880, 97, 1;
%e A172347 1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1;
%e A172347 1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508, 1;
%e A172347 1, 1159, 588772, 31940881, 442609351, 931809160, 442609351, 31940881, 588772, 1159, 1;
%t A172347 Clear[f, c, a, t];
%t A172347 f[0, a_] := 0; f[1, a_] := 1;
%t A172347 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172347 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172347 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172347 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172347 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172347 Cf. A010048 (m=1), A015109 (m=2), A172349 (m=4).
%K A172347 nonn,tabl
%O A172347 0,8
%A A172347 _Roger L. Bagula_ and _Gary W. Adamson_, Feb 01 2010