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A210553 Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.

%I A210553 #5 Mar 30 2012 18:58:16
%S A210553 1,2,1,3,2,2,4,3,5,3,5,4,9,8,5,6,5,14,15,15,8,7,6,20,24,31,26,13,8,7,
%T A210553 27,35,54,57,46,21,9,8,35,48,85,104,108,80,34,10,9,44,63,125,170,209,
%U A210553 199,139,55,11,10,54,80,175,258,360,404,366,240,89,12,11,65,99
%N A210553 Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.
%C A210553 Let T(n,k) denote the term in row n, column k.
%C A210553 T(n,n):  A000045 (Fibonacci numbers)
%C A210553 T(n,n-1): A006367
%C A210553 T(n,n-2): A105423
%C A210553 T(n,1): 1,2,3,4,5,6,7,8,9,...
%C A210553 T(n,2): 1,2,3,4,5,6,7,8,9,...
%C A210553 T(n,3): A000096
%C A210553 T(n,4): A005563
%C A210553 T(n,5): A055831
%C A210553 T(n,6): A111694
%C A210553 Row sums: A000225
%C A210553 Alternating row sums: A052551
%C A210553 For a discussion and guide to related arrays, see A208510.
%F A210553 u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A210553 v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A210553 where u(1,x)=1, v(1,x)=1.
%e A210553 First five rows:
%e A210553 1
%e A210553 2...1
%e A210553 3...2...2
%e A210553 4...3...5...3
%e A210553 5...4...9...8...5
%e A210553 First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.
%t A210553 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210553 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210553 v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210553 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210553 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210553 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210553 TableForm[cu]
%t A210553 Flatten[%]   (* A210552 *)
%t A210553 Table[Expand[v[n, x]], {n, 1, z}]
%t A210553 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210553 TableForm[cv]
%t A210553 Flatten[%]   (* A210553 *)
%t A210553 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210553 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210553 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
%t A210553 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)
%Y A210553 Cf. A210552, A208510.
%K A210553 nonn,tabl
%O A210553 1,2
%A A210553 _Clark Kimberling_, Mar 22 2012