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.

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, 65, 1, 1, 181, 11765, 68237, 68237, 11765, 181, 1, 1, 441, 79821, 1037673, 3343613, 1037673, 79821, 441, 1, 1, 1165, 513765, 18598293, 134321005, 134321005

Offset: 0

Author

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

Keywords

• nonn
• tabl

Comments

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)).

Row sums are 1, 2, 3, 12, 65, 582, 7295, 160368, 5579485, 306868458, 26280601275,...

Examples

			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, 65, 1;
1, 181, 11765, 68237, 68237, 11765, 181, 1;
1, 441, 79821, 1037673, 3343613, 1037673, 79821, 441, 1;
1, 1165, 513765, 18598293, 134321005, 134321005, 18598293, 513765, 1165, 1;
1, 2929, 3412285, 300963537, 6052711133, 13566421505, 6052711133, 300963537, 3412285, 2929, 1;
		

Crossrefs

Cf. A010048 (m=1), A015109 (m=2), A172347 (m=3), A172350 (m=5).

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}]