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.

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, 40, 1, 1, 97, 3880, 18430, 18430, 3880, 97, 1, 1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1, 1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Keywords

• nonn
• tabl

Comments

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)).

Row sums are 1, 2, 3, 10, 44, 306, 2932, 44816, 1034844, 36119056, 1882089488,...

Examples

			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, 40, 1;
1, 97, 3880, 18430, 18430, 3880, 97, 1;
1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1;
1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508, 1;
1, 1159, 588772, 31940881, 442609351, 931809160, 442609351, 31940881, 588772, 1159, 1;
		

Crossrefs

Cf. A010048 (m=1), A015109 (m=2), A172349 (m=4).

Programs

Mathematica

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + 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}]