A208607 - OEIS

%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