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%I #5 Mar 30 2012 18:58:15
%S 1,1,1,2,3,1,3,7,6,1,4,13,18,10,1,5,21,41,39,15,1,6,31,79,108,75,21,1,
%T 7,43,136,245,250,132,28,1,8,57,216,486,661,524,217,36,1,9,73,323,875,
%U 1497,1601,1015,338,45,1,10,91,461,1464,3031,4109,3556,1844
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209568; see the Formula section.
%C For n>1, row n begins with n and ends with 1.
%C For n>1, penultimate number in row n is (n-1)st triangular number.
%C Alternating row sums: 1,0,0,1,0,0,1,0,0,1,0,0,...
%C For a discussion and guide to related arrays, see A208510.
%F u(n,x)=x*u(n-1,x)+v(n-1,x),
%F v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%e First five rows:
%e 1
%e 1...1
%e 2...3....1
%e 3...7....6....1
%e 4...13...18...10...1
%e First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t v[n_, x_] := x*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[%] (* A209567 *)
%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[%] (* A209568 *)
%Y Cf. A209568, A208510.
%K nonn,tabl
%O 1,4
%A _Clark Kimberling_, Mar 10 2012
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