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 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 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 LINKS EXAMPLE 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; 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}] CROSSREFS Cf. A010048, A172353 Sequence in context: A046584 A046599 A100285 * A178916 A046568 A046571 Adjacent sequences:  A172352 A172353 A172354 * A172356 A172357 A172358 KEYWORD nonn,tabl,less AUTHOR Roger L. Bagula, Feb 01 2010 STATUS approved

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