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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

Table of n, a(n) for n=0..53.

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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Last modified December 11 19:09 EST 2017. Contains 295919 sequences.