A174412 A product triangle sequence based on:a=3;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}]]; t(n,m,a)=If[Floor[n/2] >= m, c(n, a)/c(n - m, a), c(n, a)/c(m, a)]

1, 1, 1, 1, 4, 1, 1, 28, 28, 1, 1, 133, 3724, 133, 1, 1, 760, 101080, 101080, 760, 1, 1, 3880, 2948800, 392190400, 2948800, 3880, 1, 1, 21049, 81670120, 62069291200, 62069291200, 81670120, 21049, 1, 1, 110236, 2320357564, 9002987348320

Offset

0,5

Comments

Row sums are:

{1, 2, 6, 58, 3992, 203682, 398095762, 124301964740, 6860281000355442,

10604146187026386826, 16491558999095964500974244,...}

Links

Table of n, a(n) for n=0..39.

Formula

a=3;

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}]];

t(n,m,a)=If[Floor[n/2] >= m, c(n, a)/c(n - m, a), c(n, a)/c(m, a)]

Example

{1},
{1, 1},
{1, 4, 1},
{1, 28, 28, 1},
{1, 133, 3724, 133, 1},
{1, 760, 101080, 101080, 760, 1},
{1, 3880, 2948800, 392190400, 2948800, 3880, 1},
{1, 21049, 81670120, 62069291200, 62069291200, 81670120, 21049, 1},
{1, 110236, 2320357564, 9002987348320, 6842270384723200, 9002987348320, 2320357564, 110236, 1},
{1, 588772, 64903870192, 1366161563671408, 5300706867045063040, 5300706867045063040, 1366161563671408, 64903870192, 588772, 1},
{1, 3109597, 1830843644884, 201824880037432624, 4248211899907919302576, 16483062171642726893994880, 4248211899907919302576, 201824880037432624, 1830843644884, 3109597, 1}

Mathematica

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}]];
t[n_, m_, q_] = If[Floor[n/2] >= m, c[n, q]/c[n - m, q], c[n, q]/c[m, q]];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]

Crossrefs

Sequence in context: A113370 A078536 A173918 * A177939 A209196 A158390

Adjacent sequences: A174409 A174410 A174411 * A174413 A174414 A174415

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 19 2010

Status

approved