cp's OEIS Frontend

A173006 A product triangle sequence based on recursion:a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a).

%I A173006 #4 Jun 02 2025 02:41:01
%S A173006 1,1,1,1,11,1,1,120,120,1,1,1309,14280,1309,1,1,14279,1699201,1699201,
%T A173006 14279,1,1,155760,202190640,2205562898,202190640,155760,1,1,1699081,
%U A173006 24058986960,2862818956682,2862818956682,24058986960,1699081,1,1
%N A173006 A product triangle sequence based on recursion:a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a).
%C A173006 Row sums are:
%C A173006 {1, 2, 13, 242, 16900, 3426962, 2610255700, 5773759285448, 47972252879976100,
%C A173006 1157507562695117906888, 104909162208463229766370000,...}.
%C A173006 a = 1; A034801.
%C A173006 a = 2; A156600.
%C A173006 a = 3; A156602.
%C A173006 This result seems to connect these new recursions directly to q-forms.
%F A173006 a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a);
%F A173006 c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A173006 t(n,m)=c(n)/(c(m)*c(n-m)
%e A173006 {1},
%e A173006 {1, 1},
%e A173006 {1, 11, 1},
%e A173006 {1, 120, 120, 1},
%e A173006 {1, 1309, 14280, 1309, 1},
%e A173006 {1, 14279, 1699201, 1699201, 14279, 1},
%e A173006 {1, 155760, 202190640, 2205562898, 202190640, 155760, 1},
%e A173006 {1, 1699081, 24058986960, 2862818956682, 2862818956682, 24058986960, 1699081, 1},
%e A173006 {1, 18534131, 2862817257601, 3715936800366098, 40534653607660438, 3715936800366098, 2862817257601, 18534131, 1},
%e A173006 {1, 202176360, 340651194667560, 4823283104057937603, 573930157592104171920, 573930157592104171920, 4823283104057937603, 340651194667560, 202176360, 1},
%e A173006 {1, 2205405829, 40534629348182040, 6260617753130421176727, 8126277060814812179812443, 88644086770258081457215920, 8126277060814812179812443, 6260617753130421176727, 40534629348182040, 2205405829, 1}
%t A173006 Clear[f, c, a, t];
%t A173006 f[0, a_] := 0; f[1, a_] := 1;
%t A173006 f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a];
%t A173006 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A173006 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A173006 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A173006 A034801, A156600., A156602.
%K A173006 nonn,tabl,uned
%O A173006 0,5
%A A173006 _Roger L. Bagula_, Feb 07 2010