_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Feb 01 2010

Second beta integer combination triangle of a Narayana type: a=3:f(n, a) = a*f(n - 1, a) + f(n - 2, a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];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))

1, 1, 1, 1, 10, 1, 1, 110, 110, 1, 1, 1199, 13189, 1199, 1, 1, 13080, 1568292, 1568292, 13080, 1, 1, 142680, 186625440, 2034217296, 186625440, 142680, 1, 1, 1556401, 22206729468, 2640582013104, 2640582013104, 22206729468, 1556401, 1, 1

1,5

Row sums are:

{1, 2, 12, 222, 15589, 3162746, 2407753538, 5325580597948, 44250590408026536,

1067713385571585752220,...}

a=3:

f(n, a) = a*f(n - 1, a) + f(n - 2, a);

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

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))

{1},

{1, 1},

{1, 10, 1},

{1, 110, 110, 1},

{1, 1199, 13189, 1199, 1},

{1, 13080, 1568292, 1568292, 13080, 1},

{1, 142680, 186625440, 2034217296, 186625440, 142680, 1},

{1, 1556401, 22206729468, 2640582013104, 2640582013104, 22206729468, 1556401, 1},

{1, 16977730, 2642415594973, 3427453246279524, 37390399050322080, 3427453246279524, 2642415594973, 16977730, 1},

{1, 185198630, 314425233650990, 4448834073451222609, 529407544286922803880, 529407544286922803880, 4448834073451222609, 314425233650990, 185198630, 1}

Clear[t, n, m, c, q, w, f, a] f[0, a_] := 0; f[1, a_] := 1;

f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];

c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];

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]);

Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 1, 12}];

Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 1, 12}]

nonn,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

approved