A172377 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))

1, 1, 1, 1, 5, 1, 1, 30, 30, 1, 1, 174, 1044, 174, 1, 1, 1015, 35322, 35322, 1015, 1, 1, 5915, 1200745, 6964321, 1200745, 5915, 1, 1, 34476, 40785108, 1379896154, 1379896154, 40785108, 34476, 1, 1, 200940, 1385521488, 273178653384

Offset

1,5

Comments

Row sums are:

{1, 2, 4, 14, 77, 682, 9570, 218220, 8079864, 483294396,...}

Links

Table of n, a(n) for n=1..40.

Formula

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))

Example

{1},
{1, 1},
{1, 5, 1},
{1, 30, 30, 1},
{1, 174, 1044, 174, 1},
{1, 1015, 35322, 35322, 1015, 1},
{1, 5915, 1200745, 6964321, 1200745, 5915, 1},
{1, 34476, 40785108, 1379896154, 1379896154, 40785108, 34476, 1},
{1, 200940, 1385521488, 273178653384, 1593542144740, 273178653384, 1385521488, 200940, 1},
{1, 1171165, 47066779020, 54089142449784, 1838719986152140, 1838719986152140, 54089142449784, 47066779020, 1171165, 1}

Mathematica

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}]

Crossrefs

Sequence in context: A118190 A172342 A143213 * A156587 A058720 A015116

Adjacent sequences: A172374 A172375 A172376 * A172378 A172379 A172380

Keyword

nonn,uned

Author

Roger L. Bagula, Feb 01 2010

Status

approved