%I #5 Mar 30 2012 18:58:15
%S 1,2,2,3,5,3,4,10,11,4,5,17,28,21,5,6,26,58,67,36,6,7,37,105,166,142,
%T 57,7,8,50,173,350,416,274,85,8,9,65,266,659,1011,940,491,121,9,10,82,
%U 388,1141,2156,2612,1955,829,166,10,11,101,543,1852,4172,6265
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209567; see the Formula section.
%C Row n begins and ends with n.
%C Column 2: (n-1)^2, for n>1.
%C Alternating row sums: 1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,...
%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 2...2
%e 3...5...3
%e 4...10...11...4
%e 5...17...28...21...5
%e First three polynomials v(n,x): 1, 2+2x, 3+5x+3x^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. A209567, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 10 2012