A210187 - OEIS

%I #6 Mar 30 2012 18:58:16

%S 1,3,6,2,10,9,2,15,25,13,2,21,55,49,17,2,28,105,140,81,21,2,36,182,

%T 336,285,121,25,2,45,294,714,825,506,169,29,2,55,450,1386,2079,1716,

%U 819,225,33,2,66,660,2508,4719,5005,3185,1240,289,37,2,78,935,4290

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210188; see the Formula section.

%C Row sums: even-indexed Fibonacci numbers: 1,3,8,21,55,...

%C Column 1:  triangular numbers: A000217

%C Alternating row sums:  periodic with period (1,3,4,3,1,0)

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x)=u(n-1,x)+v(n-1,x)+1,

%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 3

%e 6....2

%e 10...9....2

%e 15...25...13...2

%e First three polynomials u(n,x): 1, 3, 6 + 2x.

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;

%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[%]    (* A210187 *)

%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[%]    (* A210188 *)

%Y Cf. A210188, A208510.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Mar 18 2012