A172377 - OEIS

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

%S 1,1,1,1,5,1,1,30,30,1,1,174,1044,174,1,1,1015,35322,35322,1015,1,1,

%T 5915,1200745,6964321,1200745,5915,1,1,34476,40785108,1379896154,

%U 1379896154,40785108,34476,1,1,200940,1385521488,273178653384

%N Second beta integer combination triangle of a Narayana type: a=2: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, 4, 14, 77, 682, 9570, 218220, 8079864, 483294396,...}

%F a=2:

%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, 5, 1},

%e {1, 30, 30, 1},

%e {1, 174, 1044, 174, 1},

%e {1, 1015, 35322, 35322, 1015, 1},

%e {1, 5915, 1200745, 6964321, 1200745, 5915, 1},

%e {1, 34476, 40785108, 1379896154, 1379896154, 40785108, 34476, 1},

%e {1, 200940, 1385521488, 273178653384, 1593542144740, 273178653384, 1385521488, 200940, 1},

%e {1, 1171165, 47066779020, 54089142449784, 1838719986152140, 1838719986152140, 54089142449784, 47066779020, 1171165, 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