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A133567 A007318 * A133566.

%I A133567 #9 Mar 23 2022 07:45:00
%S A133567 1,1,1,1,3,1,1,6,3,1,1,10,6,5,1,1,15,10,15,5,1,1,21,15,35,15,7,1,1,28,
%T A133567 21,70,35,28,7,1,1,36,28,126,70,84,28,9,1,1,45,36,210,126,210,84,45,9,
%U A133567 1
%N A133567 A007318 * A133566.
%C A133567 Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).
%C A133567 From _Clark Kimberling_, Feb 28 2012: (Start)
%C A133567 A133567 is jointly generated with A133084 as an array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + (x+1)*v(n-1)x and v(n,x) = x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section. (End)
%F A133567 Binomial transform of triangle A133566.
%e A133567 First few rows of the triangle:
%e A133567   1;
%e A133567   1,  1;
%e A133567   1,  3,  1;
%e A133567   1,  6,  3,  1;
%e A133567   1, 10,  6,  5,  1;
%e A133567   1, 15, 10, 15,  5,  1;
%e A133567   1, 21, 15, 35, 15,  7,  1;
%e A133567   ...
%t A133567 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A133567 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A133567 v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A133567 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A133567 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A133567 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A133567 TableForm[cu]
%t A133567 Flatten[%]  (* A133567 *)
%t A133567 Table[Expand[v[n, x]], {n, 1, z}]
%t A133567 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A133567 TableForm[cv]
%t A133567 Flatten[%]  (* A133084 *)
%t A133567 (* _Clark Kimberling_, Feb 28 2012 *)
%Y A133567 Cf. A133566, A083329, A133084.
%K A133567 nonn,tabl
%O A133567 1,5
%A A133567 _Gary W. Adamson_, Sep 16 2007