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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).

%I A173005 #4 Jun 02 2025 02:40:55
%S A173005 1,1,1,1,9,1,1,80,80,1,1,711,6320,711,1,1,6319,499201,499201,6319,1,1,
%T A173005 56160,39430560,350439102,39430560,56160,1,1,499121,3114515040,
%U A173005 246007756722,246007756722,3114515040,499121,1,1,4435929,246007257601
%N A173005 A product triangle sequence based on recursion:a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a).
%C A173005 Row sums are:
%C A173005 {1, 2, 11, 162, 7744, 1011042, 429412544, 498245541768, 1880728607247424,
%C A173005 19394268001029953928, 650631110504313946320896,...}.
%C A173005 a = 1; A034801.
%C A173005 a = 2; A156600.
%C A173005 a = 3; A156602.
%C A173005 This result seems to connect these new recursions directly to q-forms.
%F A173005 a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a);
%F A173005 c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A173005 t(n,m)=c(n)/(c(m)*c(n-m)
%e A173005 {1},
%e A173005 {1, 1},
%e A173005 {1, 9, 1},
%e A173005 {1, 80, 80, 1},
%e A173005 {1, 711, 6320, 711, 1},
%e A173005 {1, 6319, 499201, 499201, 6319, 1},
%e A173005 {1, 56160, 39430560, 350439102, 39430560, 56160, 1},
%e A173005 {1, 499121, 3114515040, 246007756722, 246007756722, 3114515040, 499121, 1},
%e A173005 {1, 4435929, 246007257601, 172697094835902, 1534842394188558, 172697094835902, 246007257601, 4435929, 1},
%e A173005 {1, 39424240, 19431458835440, 121233114567545603, 9575881454449171680, 9575881454449171680, 121233114567545603, 19431458835440, 39424240, 1},
%e A173005 {1, 350382231, 1534839240742160, 85105473729326613333, 59743922859711995180563, 530973050767752120484320, 59743922859711995180563, 85105473729326613333, 1534839240742160, 350382231, 1}
%t A173005 Clear[f, c, a, t];
%t A173005 f[0, a_] := 0; f[1, a_] := 1;
%t A173005 f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a];
%t A173005 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A173005 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A173005 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A173005 A034801, A156600., A156602.
%K A173005 nonn,tabl,uned
%O A173005 0,5
%A A173005 _Roger L. Bagula_, Feb 07 2010