A172353 Triangle t(n,k) of Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c(n) = A126772(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 4, 2, 1, 1, 3, 6, 12, 6, 3, 1, 1, 4, 12, 24, 24, 12, 4, 1, 1, 5, 20, 60, 60, 60, 20, 5, 1, 1, 7, 35, 140, 210, 210, 140, 35, 7, 1, 1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1

Offset: 0

Author

Roger L. Bagula, Feb 01 2010

Keywords

• nonn
• tabl

Comments

Start from the Padovan sequence A134816 and its partial products A126772, extended by A126772(0)=1. Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 4, 8, 14, 32, 82, 232, 786, 2981,..

Examples

			1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 2, 4, 4, 2, 1;
1, 3, 6, 12, 6, 3, 1;
1, 4, 12, 24, 24, 12, 4, 1;
1, 5, 20, 60, 60, 60, 20, 5, 1;
1, 7, 35, 140, 210, 210, 140, 35, 7, 1;
1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1;
		

Crossrefs

Cf. A010048, A172355

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}]