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%I #5 Mar 30 2012 18:58:13
%S 1,3,1,5,3,1,7,8,6,1,9,18,19,6,1,11,35,47,25,9,1,13,61,102,81,42,9,1,
%T 15,98,203,219,147,51,12,1,17,148,378,520,435,216,74,12,1,19,213,666,
%U 1122,1145,747,334,86,15,1,21,295,1119,2250,2753,2233,1245,450
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208606; see the Formula section.
%C Alternating rows sums: 1,2,3,4,5,6,7,8,...
%F u(n,x)=u(n-1,x)+x*v(n-1,x),
%F v(n,x)=(x+1)*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 3...1
%e 5...3...1
%e 7...8...6...1
%e 9...18...19...6...1
%e First five polynomials v(n,x):
%e 1
%e 3 + x
%e 5 + 3x + x^2
%e 7 + 8x + 6x^2 + x^3
%e 9 + 18x + 19x^2 + 6x^3 + x^4
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := (x + 1)*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[%] (* A208606 *)
%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[%] (* A208607 *)
%Y Cf. A208606.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Feb 29 2012
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