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Triangle t(n,k) of Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c(n) = A126772(n).
4

%I #5 Jul 05 2012 12:06:32

%S 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,

%T 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,

%U 63,315,630,945,630,315,63,9,1

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

%C 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)).

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

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 1, 1, 1;

%e 1, 2, 2, 2, 1;

%e 1, 2, 4, 4, 2, 1;

%e 1, 3, 6, 12, 6, 3, 1;

%e 1, 4, 12, 24, 24, 12, 4, 1;

%e 1, 5, 20, 60, 60, 60, 20, 5, 1;

%e 1, 7, 35, 140, 210, 210, 140, 35, 7, 1;

%e 1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1;

%t Clear[f, c, a, t];

%t f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;

%t f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, a];

%t c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];

%t t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);

%t Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];

%t Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

%Y Cf. A010048, A172355

%K nonn,tabl

%O 0,12

%A _Roger L. Bagula_, Feb 01 2010