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

%I A210552 #7 Mar 30 2012 18:58:16
%S A210552 1,1,2,1,3,3,1,4,5,5,1,5,7,10,8,1,6,9,16,18,13,1,7,11,23,31,33,21,1,8,
%T A210552 13,31,47,62,59,34,1,9,15,40,66,101,119,105,55,1,10,17,50,88,151,205,
%U A210552 227,185,89,1,11,19,61,113,213,321,414,426,324,144,1,12,21,73
%N A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.
%C A210552 Let T(n,k) denote the term in row n, column k.
%C A210552 T(n,n):  A000045 (Fibonacci numbers)
%C A210552 T(n,n-1): A010049 (second-order Fibonacci numbers)
%C A210552 T(n,1): 1,1,1,1,1,1,1,1,1,1,1,,...
%C A210552 T(n,2): 2,3,4,5,6,7,8,9,10,11,...
%C A210552 T(n,3): 3,5,7,9,11,13,15,17,19,...
%C A210552 T(n,4): A052905
%C A210552 Row sums: A000225
%C A210552 Alternating row sums: A094024 (signed)
%C A210552 For a discussion and guide to related arrays, see A208510.
%F A210552 u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A210552 v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A210552 where u(1,x)=1, v(1,x)=1.
%e A210552 First five rows:
%e A210552 1
%e A210552 1...2
%e A210552 1...3...3
%e A210552 1...4...5...5
%e A210552 1...5...7...10...8
%e A210552 First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.
%t A210552 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210552 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210552 v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210552 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210552 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210552 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210552 TableForm[cu]
%t A210552 Flatten[%]   (* A210552 *)
%t A210552 Table[Expand[v[n, x]], {n, 1, z}]
%t A210552 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210552 TableForm[cv]
%t A210552 Flatten[%]   (* A210553 *)
%t A210552 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210552 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210552 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
%t A210552 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)
%Y A210552 Cf. A210553, A208510.
%K A210552 nonn,tabl
%O A210552 1,3
%A A210552 _Clark Kimberling_, Mar 22 2012