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A209583 Triangle of coefficients of polynomials u(n,x) jointly generated with A209584; see the Formula section.

%I A209583 #8 Mar 30 2012 18:58:15
%S A209583 1,1,1,3,3,1,5,10,6,1,9,23,24,10,1,15,51,71,49,15,1,25,104,188,178,90,
%T A209583 21,1,41,205,452,552,390,153,28,1,67,391,1025,1530,1396,776,245,36,1,
%U A209583 109,730,2218,3927,4394,3169,1435,374,45,1,177,1339,4636,9502
%N A209583 Triangle of coefficients of polynomials u(n,x) jointly generated with A209584; see the Formula section.
%C A209583 Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,...
%C A209583 For a discussion and guide to related arrays, see A208510.
%F A209583 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209583 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A209583 where u(1,x)=1, v(1,x)=1.
%e A209583 First five rows:
%e A209583 1
%e A209583 1...1
%e A209583 3...3....1
%e A209583 5...10...6....1
%e A209583 9...23...24...10...1
%e A209583 First three polynomials u(n,x): 1, 1 + x, 3 + 3x + x^2.
%t A209583 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209583 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209583 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209583 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209583 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209583 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209583 TableForm[cu]
%t A209583 Flatten[%]   (* A209583 *)
%t A209583 Table[Expand[v[n, x]], {n, 1, z}]
%t A209583 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209583 TableForm[cv]
%t A209583 Flatten[%]   (* A209584 *)
%Y A209583 Cf. A209584, A208510.
%K A209583 nonn,tabl
%O A209583 1,4
%A A209583 _Clark Kimberling_, Mar 11 2012