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%I #6 Mar 30 2012 18:58:15
%S 1,1,2,2,4,3,3,7,8,5,4,11,17,17,8,5,16,31,41,33,13,6,22,51,83,91,63,
%T 21,7,29,78,150,205,195,117,34,8,37,113,250,406,483,403,214,55,9,46,
%U 157,392,734,1039,1091,812,386,89,10,56,211,586,1239,2023,2536
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209756; see the Formula section.
%C Column 1: 1,2,3,4,5,6,....... A000027
%C Column 2: 1,2,4,7,11,........ A000124
%C Column 3: 2,6,13,24,......... A105163
%C Final row terms: 1,2,3,5,.... A000045 (Fibonacci numbers)
%C Row sums: 1,3,9,23,57,139,... A133654
%C Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...; A033999
%C For a discussion and guide to related arrays, see A208510.
%F u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%e First five rows:
%e 1
%e 1...2
%e 2...4....3
%e 3...7....8....5
%e 4...11...11...17...8
%e First three polynomials u(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A209755 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A209756 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A133654 *)
%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A001333 *)
%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A109613 *)
%Y Cf. A209756, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 14 2012
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