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 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)) 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are: {1, 2, 4, 14, 77, 682, 9570, 218220, 8079864, 483294396,...} LINKS 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

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Last modified January 21 10:26 EST 2020. Contains 331105 sequences. (Running on oeis4.)