A174411 A product triangle sequence based on:a=1;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, 2, 1, 1, 6, 6, 1, 1, 15, 90, 15, 1, 1, 40, 600, 600, 40, 1, 1, 104, 4160, 62400, 4160, 104, 1, 1, 273, 28392, 1135680, 1135680, 28392, 273, 1, 1, 714, 194922, 20271888, 810875520, 20271888, 194922, 714, 1, 1, 1870, 1335180, 364504140

Offset

0,5

Comments

Row sums are:

{1, 2, 4, 14, 122, 1282, 70930, 2328692, 851810570, 76548543502,

189143352851092,...}

Links

Table of n, a(n) for n=0..48.

Formula

a=1;

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, 2, 1},
{1, 6, 6, 1},
{1, 15, 90, 15, 1},
{1, 40, 600, 600, 40, 1},
{1, 104, 4160, 62400, 4160, 104, 1},
{1, 273, 28392, 1135680, 1135680, 28392, 273, 1},
{1, 714, 194922, 20271888, 810875520, 20271888, 194922, 714, 1},
{1, 1870, 1335180, 364504140, 37908430560, 37908430560, 364504140, 1335180, 1870, 1},
{1, 4895, 9153650, 6535706100, 1784247765300, 185561767591200, 1784247765300, 6535706100, 9153650, 4895, 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: A173881 A329228 A172373 * A322620 A155795 A009963

Adjacent sequences: A174408 A174409 A174410 * A174412 A174413 A174414

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 19 2010

Status

approved