A172343 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=6.

1, 1, 1, 1, 6, 1, 1, 37, 37, 1, 1, 228, 1406, 228, 1, 1, 1405, 53390, 53390, 1405, 1, 1, 8658, 2027415, 12493260, 2027415, 8658, 1, 1, 53353, 76988379, 2923477635, 2923477635, 76988379, 53353, 1, 1, 328776, 2923530988, 684106251192

Offset: 0

Author

Roger L. Bagula, Feb 01 2010

Keywords

• nonn
• tabl

Comments

Start from the generalized Fibonacci sequence A005668 and its partial products c(n) = 1, 1, 6, 222, 50616, 71115480, 615717825840, 32850393162041520... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 8, 76, 1864, 109592, 16565408, 6001038736, 5589714971584,

12478331908166432, 71624411004755875328,...

Examples

			1;
1, 1;
1, 6, 1;
1, 37, 37, 1;
1, 228, 1406, 228, 1;
1, 1405, 53390, 53390, 1405, 1;
1, 8658, 2027415, 12493260, 2027415, 8658, 1;
1, 53353, 76988379, 2923477635, 2923477635, 76988379, 53353, 1;
1, 328776, 2923530988, 684106251192, 4215654749670, 684106251192, 2923530988, 328776, 1;
		

Crossrefs

Cf. A010048 (m=1), A099927 (m=2), A172342 (m=5), A172345 (m=7).

Programs

Mathematica

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]