A172355 Triangle t(n,k) read by rows: generalized Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Padovan sequence with multiplier m=5.

1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 6, 30, 6, 1, 1, 26, 156, 156, 26, 1, 1, 35, 910, 1092, 910, 35, 1, 1, 136, 4760, 24752, 24752, 4760, 136, 1, 1, 201, 27336, 191352, 829192, 191352, 27336, 201, 1, 1, 715, 143715, 3909048, 22802780, 22802780, 3909048, 143715, 715

Offset: 0

Author

Roger L. Bagula, Feb 01 2010

Keywords

• nonn
• tabl
• less

Comments

Start from the generalized Padovan sequence f(n) = 0, 1, 1, 5, 6, 26, 35, 136, 201, 715, 1141, 3776,.. , f(n) = 5*f(n-2)+f(n-3), and its partial products c(n) = 1, 1, 1, 5, 30, 780, 27300, 3712800, 746272800, 533585052000.. Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 12, 44, 366, 2984, 59298, 1266972, 53712518, 2554657926,....

Note that rows n>= 14 contain fractions. - R. J. Mathar, Jul 05 2012

Examples

			1;
1, 1;
1, 1, 1;
1, 5, 5, 1;
1, 6, 30, 6, 1;
1, 26, 156, 156, 26, 1;
1, 35, 910, 1092, 910, 35, 1;
1, 136, 4760, 24752, 24752, 4760, 136, 1;
1, 201, 27336, 191352, 829192, 191352, 27336, 201, 1;
1, 715, 143715, 3909048, 22802780, 22802780, 3909048, 143715, 715, 1;
1, 1141, 815815, 32795763, 743370628, 1000691230, 743370628, 32795763, 815815, 1141, 1;
		

Crossrefs

Cf. A010048, A172353

Programs

Mathematica

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, 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}]