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A209756 Triangle of coefficients of polynomials v(n,x) jointly generated with A209755; see the Formula section.

%I A209756 #8 Mar 30 2012 18:58:15
%S A209756 1,2,1,3,2,2,4,4,6,3,5,7,13,11,5,6,11,24,28,22,8,7,16,40,59,63,41,13,
%T A209756 8,22,62,110,146,132,76,21,9,29,91,188,296,337,271,138,34,10,37,128,
%U A209756 301,546,743,754,541,248,55,11,46,174,458,938,1477,1793,1632
%N A209756 Triangle of coefficients of polynomials v(n,x) jointly generated with A209755; see the Formula section.
%C A209756 Column 1: 1,2,3,4,5,6,....... A000027
%C A209756 Column 2: 1,2,4,7,11,........ A000124
%C A209756 Column 3: 2,6,13,24,......... A003600
%C A209756 Last term in row n: 1,1,2,3,. A000045 (Fibonacci numbers)
%C A209756 Row sums:  1,3,7,17,41,...... A001333
%C A209756 Alternating row sums: 1,2,3,3,5,5,7,7,...; A109613
%C A209756 For a discussion and guide to related arrays, see A208510.
%F A209756 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209756 v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A209756 where u(1,x)=1, v(1,x)=1.
%e A209756 First five rows:
%e A209756 1
%e A209756 2...1
%e A209756 3...2...2
%e A209756 4...4...6....3
%e A209756 5...7...13...11...5
%e A209756 First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.
%t A209756 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209756 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209756 v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A209756 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209756 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209756 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209756 TableForm[cu]
%t A209756 Flatten[%]    (* A209755 *)
%t A209756 Table[Expand[v[n, x]], {n, 1, z}]
%t A209756 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209756 TableForm[cv]
%t A209756 Flatten[%]    (* A209756 *)
%t A209756 Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A133654 *)
%t A209756 Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A001333 *)
%t A209756 Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A033999 *)
%t A209756 Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A109613 *)
%Y A209756 Cf. A208510.
%K A209756 nonn,tabl
%O A209756 1,2
%A A209756 _Clark Kimberling_, Mar 14 2012