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%I #9 Mar 23 2022 07:45:00
%S 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 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 1
%N A007318 * A133566.
%C Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).
%C From _Clark Kimberling_, Feb 28 2012: (Start)
%C 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 Binomial transform of triangle A133566.
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 3, 1;
%e 1, 10, 6, 5, 1;
%e 1, 15, 10, 15, 5, 1;
%e 1, 21, 15, 35, 15, 7, 1;
%e ...
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A133567 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A133084 *)
%t (* _Clark Kimberling_, Feb 28 2012 *)
%Y Cf. A133566, A083329, A133084.
%K nonn,tabl
%O 1,5
%A _Gary W. Adamson_, Sep 16 2007
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