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 A172376 A beta integer combination triangle of a Narayana type: a=3:f(n, a) = a*f(n - 1, a) + 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, 12, 1, 1, 180, 180, 1, 1, 2565, 38475, 2565, 1, 1, 36936, 7895070, 7895070, 36936, 1, 1, 530712, 1633531536, 23277824388, 1633531536, 530712, 1, 1, 7628985, 337399490610, 69234375473172, 69234375473172, 337399490610, 7628985, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are: {1, 2, 14, 362, 43607, 15864014, 26545948886, 139143565185536, 3371062783875324716, 253833990333055824621332,...} LINKS FORMULA a=3: f(n, a) = a*f(n - 1, a) + 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, 12, 1}, {1, 180, 180, 1}, {1, 2565, 38475, 2565, 1}, {1, 36936, 7895070, 7895070, 36936, 1}, {1, 530712, 1633531536, 23277824388, 1633531536, 530712, 1}, {1, 7628985, 337399490610, 69234375473172, 69234375473172, 337399490610, 7628985, 1}, {1, 109656180, 69713779364775, 205544107079102610, 2959835141939077584, 205544107079102610, 69713779364775, 109656180, 1}, {1, 1576188396, 14403233205473940, 610455833755903367505, 126306524929537227280824, 126306524929537227280824, 610455833755903367505, 14403233205473940, 1576188396, 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] + 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: A022175 A176627 A015129 * A289673 A010211 A066786 Adjacent sequences:  A172373 A172374 A172375 * A172377 A172378 A172379 KEYWORD nonn,uned AUTHOR Roger L. Bagula, Feb 01 2010 STATUS approved

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Last modified January 22 10:25 EST 2020. Contains 331144 sequences. (Running on oeis4.)