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A209573 Triangle of coefficients of polynomials u(n,x) jointly generated with A209574; see the Formula section.

%I A209573 #6 Feb 27 2013 09:44:12
%S A209573 1,1,1,2,4,1,3,9,9,1,4,17,29,16,1,5,28,69,74,25,1,6,42,138,224,160,36,
%T A209573 1,7,59,245,541,613,307,49,1,8,79,399,1127,1781,1469,539,64,1,9,102,
%U A209573 609,2111,4331,5103,3171,884,81,1,10,128,884,3649,9281,14419
%N A209573 Triangle of coefficients of polynomials u(n,x) jointly generated with A209574; see the Formula section.
%C A209573 For n>1, let r(n,k) be the k-th number in row n.  Then
%C A209573 r(n,1)=n-1, r(n,n-1)=(n-1)^2, and r(n,n)=1.  For a
%C A209573 discussion and guide to related arrays, see A208510.
%F A209573 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209573 v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A209573 where u(1,x)=1, v(1,x)=1.
%e A209573 First five rows:
%e A209573 1
%e A209573 1...1
%e A209573 2...4....1
%e A209573 3...9....9....1
%e A209573 4...17...29...16...1
%e A209573 First three polynomials v(n,x): 1, 1 + x, 2 + 4x + x^2.
%t A209573 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209573 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209573 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209573 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209573 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209573 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209573 TableForm[cu]
%t A209573 Flatten[%]   (* A209573 *)
%t A209573 Table[Expand[v[n, x]], {n, 1, z}]
%t A209573 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209573 TableForm[cv]
%t A209573 Flatten[%]   (* A209574 *)
%Y A209573 Cf. A209574, A208510.
%K A209573 nonn,tabl
%O A209573 1,4
%A A209573 _Clark Kimberling_, Mar 11 2012