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A209819 Triangle of coefficients of polynomials u(n,x) jointly generated with A209820; see the Formula section.

%I A209819 #6 Mar 30 2012 18:58:16
%S A209819 1,1,3,1,5,7,1,5,17,17,1,5,21,53,41,1,5,21,81,157,99,1,5,21,89,289,
%T A209819 449,239,1,5,21,89,361,973,1253,577,1,5,21,89,377,1389,3133,3433,1393,
%U A209819 1,5,21,89,377,1565,5085,9745,9273,3363,1,5,21,89,377,1597,6285
%N A209819 Triangle of coefficients of polynomials u(n,x) jointly generated with A209820; see the Formula section.
%C A209819 Let T(n,k) be the general term.
%C A209819 T(n,n): A001333
%C A209819 T(n,n-1): A088210
%C A209819 Row sums: A003561
%C A209819 Alternating row sums: 1,-2,3,-4,5,-6,7,-8,...
%C A209819 Limiting row: F(2), F(5),F(8),...where F=A000045 (Fibonacci numbers)
%C A209819 For a discussion and guide to related arrays, see A208510.
%F A209819 u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A209819 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A209819 where u(1,x)=1, v(1,x)=1.
%e A209819 First five rows:
%e A209819 1
%e A209819 1...3
%e A209819 1...5...7
%e A209819 1...5...17...17
%e A209819 1...5...21...53...41
%e A209819 First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 7x^2.
%t A209819 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209819 u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209819 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209819 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209819 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209819 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209819 TableForm[cu]
%t A209819 Flatten[%]    (* A209819 *)
%t A209819 Table[Expand[v[n, x]], {n, 1, z}]
%t A209819 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209819 TableForm[cv]
%t A209819 Flatten[%]    (* A209820 *)
%Y A209819 Cf. A209820, A208510.
%K A209819 nonn,tabl
%O A209819 1,3
%A A209819 _Clark Kimberling_, Mar 23 2012