A210204 - OEIS

%I #6 Jan 14 2022 21:25:09

%S 1,3,2,7,8,2,15,24,12,2,31,64,48,16,2,63,160,160,80,20,2,127,384,480,

%T 320,120,24,2,255,896,1344,1120,560,168,28,2,511,2048,3584,3584,2240,

%U 896,224,32,2,1023,4608,9216,10752,8064,4032,1344,288,36,2,2047

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210203; see the Formula section.

%C Column 1:  -1+2^n.

%C Row sums: A048473.

%C Alternating row sums: 1,1,1,1,1,1,1,1,1,...

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

%C Row sums without first column give A056182. - _Alois P. Heinz_, Jan 14 2022

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

%F v(n,x)=(x+1)*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....2

%e 7....8....2

%e 15...24...12...2

%e 31...64...48...16...2

%e First three polynomials v(n,x): 1, 3 + 2x , 7 + 8x + 2x^2.

%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 + 1)*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[%]   (* A210203 *)

%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[%]   (* A210204 *)

%Y Cf. A210203, A208510.

%Y Cf. A056182.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 18 2012