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%I #5 Mar 31 2012 20:30:51
%S 1,1,1,1,2,2,1,3,3,3,1,4,6,5,5,1,5,8,12,8,8,1,6,12,17,23,13,13,1,7,15,
%T 29,33,43,21,21,1,8,20,38,64,63,79,34,34,1,9,24,56,86,136,117,143,55,
%U 55,1,10,30,70,140,187,279,214,256,89,89,1,11,35,95,180,332
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210871; see the Formula section.
%C In row n, for n>1, the first two terms are 1 and n-1, and the last two are F(n) and F(n), where F = A000045 (Fibonacci numbers).
%C Row sums: A000975
%C Alternating row sums: A113954
%C For a discussion and guide to related arrays, see A208510.
%F u(n,x)=u(n-1,x)+x*v(n-1,x),
%F v(n,x)=(x+1)*u(n-1,x)+(x-1)*v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%e First six rows:
%e 1
%e 1...1
%e 1...2...2
%e 1...3...3...3
%e 1...4...6...5....5
%e 1...5...8...12...8...8
%e First three polynomials u(n,x): 1, 1 + x, 1 + 2x + 2x^2.
%t u[1, x_] := 1; v[1, x_] := 1; z = 14;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := (x + 1)*u[n - 1, x] + (x - 1)*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[%] (* A210870 *)
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A210871 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000975 *)
%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A001045 *)
%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A113954 *)
%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
%Y Cf. A210871, A208510.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Mar 29 2012
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