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A209571 Triangle of coefficients of polynomials u(n,x) jointly generated with A209572; see the Formula section.

%I A209571 #5 Mar 30 2012 18:58:15
%S A209571 1,1,1,1,4,1,1,4,9,1,1,4,15,16,1,1,4,15,44,25,1,1,4,15,56,105,36,1,1,
%T A209571 4,15,56,185,216,49,1,1,4,15,56,209,524,399,64,1,1,4,15,56,209,732,
%U A209571 1295,680,81,1,1,4,15,56,209,780,2303,2864,1089,100,1,1,4,15,56
%N A209571 Triangle of coefficients of polynomials u(n,x) jointly generated with A209572; see the Formula section.
%C A209571 Penultimate number in row n is (n-1)^2, for n>1.
%C A209571 Combinatorial limit of row n satisfies linear recurrence
%C A209571 r(n)=4*r(n-1)-r(n-2) with r(1)=1 and r(2)=4. For a
%C A209571 discussion and guide to related arrays, see A208510.
%F A209571 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209571 v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
%F A209571 where u(1,x)=1, v(1,x)=1.
%e A209571 First five rows:
%e A209571 1
%e A209571 1...1
%e A209571 1...4....1
%e A209571 1...4....9....1
%e A209571 1...4....15...16...1
%e A209571 First three polynomials v(n,x): 1, 1 + x, 1 + 4x + x^2.
%t A209571 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209571 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209571 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209571 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209571 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209571 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209571 TableForm[cu]
%t A209571 Flatten[%]   (* A209571 *)
%t A209571 Table[Expand[v[n, x]], {n, 1, z}]
%t A209571 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209571 TableForm[cv]
%t A209571 Flatten[%]   (* A209572 *)
%Y A209571 Cf. A209572, A208510.
%K A209571 nonn,tabl
%O A209571 1,5
%A A209571 _Clark Kimberling_, Mar 11 2012