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A209567 Triangle of coefficients of polynomials u(n,x) jointly generated with A209568; see the Formula section.

%I A209567 #5 Mar 30 2012 18:58:15
%S A209567 1,1,1,2,3,1,3,7,6,1,4,13,18,10,1,5,21,41,39,15,1,6,31,79,108,75,21,1,
%T A209567 7,43,136,245,250,132,28,1,8,57,216,486,661,524,217,36,1,9,73,323,875,
%U A209567 1497,1601,1015,338,45,1,10,91,461,1464,3031,4109,3556,1844
%N A209567 Triangle of coefficients of polynomials u(n,x) jointly generated with A209568; see the Formula section.
%C A209567 For n>1, row n begins with n and ends with 1.
%C A209567 For n>1, penultimate number in row n is (n-1)st triangular number.
%C A209567 Alternating row sums:  1,0,0,1,0,0,1,0,0,1,0,0,...
%C A209567 For a discussion and guide to related arrays, see A208510.
%F A209567 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209567 v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A209567 where u(1,x)=1, v(1,x)=1.
%e A209567 First five rows:
%e A209567 1
%e A209567 1...1
%e A209567 2...3....1
%e A209567 3...7....6....1
%e A209567 4...13...18...10...1
%e A209567 First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.
%t A209567 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209567 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209567 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209567 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209567 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209567 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209567 TableForm[cu]
%t A209567 Flatten[%]   (* A209567 *)
%t A209567 Table[Expand[v[n, x]], {n, 1, z}]
%t A209567 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209567 TableForm[cv]
%t A209567 Flatten[%]   (* A209568 *)
%Y A209567 Cf. A209568, A208510.
%K A209567 nonn,tabl
%O A209567 1,4
%A A209567 _Clark Kimberling_, Mar 10 2012