A174410 A product triangle sequence based on:a=2;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, 3, 1, 1, 15, 15, 1, 1, 55, 825, 55, 1, 1, 231, 12705, 12705, 231, 1, 1, 903, 208593, 11472615, 208593, 903, 1, 1, 3655, 3300465, 762407415, 762407415, 3300465, 3655, 1, 1, 14535, 53125425, 47972258775, 11081591777025, 47972258775

Offset

0,5

Comments

Row sums are:

{1, 2, 5, 32, 937, 25874, 11891609, 1531423072, 11177642574497,

5600818051389794, 652945346856872982409,...}

Links

Table of n, a(n) for n=0..41.

Formula

a=2;

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, 3, 1},
{1, 15, 15, 1},
{1, 55, 825, 55, 1},
{1, 231, 12705, 12705, 231, 1},
{1, 903, 208593, 11472615, 208593, 903, 1},
{1, 3655, 3300465, 762407415, 762407415, 3300465, 3655, 1},
{1, 14535, 53125425, 47972258775, 11081591777025, 47972258775, 53125425, 14535, 1},
{1, 58311, 847550385, 3097796657175, 2797310381429025, 2797310381429025, 3097796657175, 847550385, 58311, 1},
{1, 232903, 13580806833, 197397027317655, 721486134846029025, 651501979765964209575, 721486134846029025, 197397027317655, 13580806833, 232903, 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: A326800 A176225 A173917 * A156690 A228900 A060325

Adjacent sequences: A174407 A174408 A174409 * A174411 A174412 A174413

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 19 2010

Status

approved