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A209581 Triangle of coefficients of polynomials u(n,x) jointly generated with A209582; see the Formula section.

%I A209581 #6 Mar 30 2012 18:58:15
%S A209581 1,1,1,2,4,1,2,8,11,1,3,12,25,26,1,3,19,51,71,57,1,4,24,88,180,190,
%T A209581 120,1,4,34,136,348,566,486,247,1,5,40,206,628,1238,1648,1199,502,1,5,
%U A209581 53,282,1034,2526,4070,4545,2873,1013,1,6,60,395,1600,4556,9208
%N A209581 Triangle of coefficients of polynomials u(n,x) jointly generated with A209582; see the Formula section.
%C A209581 For a discussion and guide to related arrays, see A208510.
%F A209581 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209581 v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
%F A209581 where u(1,x)=1, v(1,x)=1.
%e A209581 First five rows:
%e A209581 1
%e A209581 1...1
%e A209581 2...4....1
%e A209581 2...8....11...1
%e A209581 3...12...25...26...1
%e A209581 First three polynomials v(n,x): 1, 1 + x, 2 + 4x + x^2.
%t A209581 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209581 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209581 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209581 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209581 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209581 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209581 TableForm[cu]
%t A209581 Flatten[%]   (* A209581 *)
%t A209581 Table[Expand[v[n, x]], {n, 1, z}]
%t A209581 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209581 TableForm[cv]
%t A209581 Flatten[%]   (* A209582 *)
%Y A209581 Cf. A209582, A208510.
%K A209581 nonn,tabl
%O A209581 1,4
%A A209581 _Clark Kimberling_, Mar 11 2012