A173918 A double product sequence based on a=3;f(n,a)=f(n-1,a)+a*f(n-2,a)

1, 1, 1, 1, 4, 1, 1, 28, 28, 1, 1, 133, 931, 133, 1, 1, 760, 25270, 25270, 760, 1, 1, 3880, 737200, 3501700, 737200, 3880, 1, 1, 21049, 20417530, 554190100, 554190100, 20417530, 21049, 1, 1, 110236, 580089391, 80383815610, 459336089200

Offset

0,5

Comments

Row sums are:

{1, 2, 6, 58, 1199, 52062, 4983862, 1149257360, 621264119676, 736123531927540,

2029965262731727157,...}.

I get these same sequences using a Narayana form:

w[n_, m_, q_] := c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*f[m, q])

Links

Table of n, a(n) for n=0..40.

Formula

c(n,a)=If[n == 0, 1, Product[f(i, a)*f(i + 1, a), {i, 1, n}]]

Example

{1},
{1, 1},
{1, 4, 1},
{1, 28, 28, 1},
{1, 133, 931, 133, 1},
{1, 760, 25270, 25270, 760, 1},
{1, 3880, 737200, 3501700, 737200, 3880, 1},
{1, 21049, 20417530, 554190100, 554190100, 20417530, 21049, 1},
{1, 110236, 580089391, 80383815610, 459336089200, 80383815610, 580089391, 110236, 1},
{1, 588772, 16225967548, 12197871104209, 355847668303240, 355847668303240, 12197871104209, 16225967548, 588772, 1},
{1, 3109597, 457710911221, 1802007857477077, 285191454075451081, 1455977423437829203, 285191454075451081, 1802007857477077, 457710911221, 3109597, 1}

Mathematica

Clear[t, n, m, c, q, w, f, a];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a]*f[i + 1, a], {i, 1, n}]];
w[n_, m_, q_] := c[n, q]/(c[m, q]*c[n - m, q]);
Table[Table[Table[w[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 12}];
Table[Flatten[Table[Table[w[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 12}]

Crossrefs

Sequence in context: A140805 A113370 A078536 * A174412 A177939 A209196

Adjacent sequences: A173915 A173916 A173917 * A173919 A173920 A173921

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 02 2010

Status

approved