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 A173005 A product triangle sequence based on recursion:a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a) 0
 1, 1, 1, 1, 9, 1, 1, 80, 80, 1, 1, 711, 6320, 711, 1, 1, 6319, 499201, 499201, 6319, 1, 1, 56160, 39430560, 350439102, 39430560, 56160, 1, 1, 499121, 3114515040, 246007756722, 246007756722, 3114515040, 499121, 1, 1, 4435929, 246007257601 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, 11, 162, 7744, 1011042, 429412544, 498245541768, 1880728607247424, 19394268001029953928, 650631110504313946320896,...}. a = 1; A034801. a = 2; A156600. a = 3; A156602. This result seems to connect these new recursions directly to q-forms. LINKS FORMULA a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a); c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]]; t(n,m)=c(n)/(c(m)*c(n-m) EXAMPLE {1}, {1, 1}, {1, 9, 1}, {1, 80, 80, 1}, {1, 711, 6320, 711, 1}, {1, 6319, 499201, 499201, 6319, 1}, {1, 56160, 39430560, 350439102, 39430560, 56160, 1}, {1, 499121, 3114515040, 246007756722, 246007756722, 3114515040, 499121, 1}, {1, 4435929, 246007257601, 172697094835902, 1534842394188558, 172697094835902, 246007257601, 4435929, 1}, {1, 39424240, 19431458835440, 121233114567545603, 9575881454449171680, 9575881454449171680, 121233114567545603, 19431458835440, 39424240, 1}, {1, 350382231, 1534839240742160, 85105473729326613333, 59743922859711995180563, 530973050767752120484320, 59743922859711995180563, 85105473729326613333, 1534839240742160, 350382231, 1} MATHEMATICA Clear[f, c, a, t]; f[0, a_] := 0; f[1, a_] := 1; f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a]; c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]]; t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]); Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}] CROSSREFS A034801, A156600., A156602. Sequence in context: A202988 A098436 A022172 * A015123 A176647 A068452 Adjacent sequences:  A173002 A173003 A173004 * A173006 A173007 A173008 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, Feb 07 2010 STATUS approved

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Last modified October 1 13:04 EDT 2020. Contains 337443 sequences. (Running on oeis4.)