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A172378
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))
0
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
OFFSET
1,5
COMMENTS
Row sums are:
{1, 2, 12, 222, 15589, 3162746, 2407753538, 5325580597948, 44250590408026536,
1067713385571585752220,...}
FORMULA
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))
EXAMPLE
{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}
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: A176243 A022173 A158117 * A015124 A156767 A365025
KEYWORD
nonn,uned
AUTHOR
Roger L. Bagula, Feb 01 2010
STATUS
approved