A172378 - OEIS

%I #2 Mar 30 2012 17:34:38

%S 1,1,1,1,10,1,1,110,110,1,1,1199,13189,1199,1,1,13080,1568292,1568292,

%T 13080,1,1,142680,186625440,2034217296,186625440,142680,1,1,1556401,

%U 22206729468,2640582013104,2640582013104,22206729468,1556401,1,1

%N 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))

%C Row sums are:

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

%C 1067713385571585752220,...}

%F a=3:

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

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

%F 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))

%e {1},

%e {1, 1},

%e {1, 10, 1},

%e {1, 110, 110, 1},

%e {1, 1199, 13189, 1199, 1},

%e {1, 13080, 1568292, 1568292, 13080, 1},

%e {1, 142680, 186625440, 2034217296, 186625440, 142680, 1},

%e {1, 1556401, 22206729468, 2640582013104, 2640582013104, 22206729468, 1556401, 1},

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

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

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

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

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

%t 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]);

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

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

%K nonn,uned

%O 1,5

%A _Roger L. Bagula_, Feb 01 2010