A210551 - OEIS

%I #25 May 18 2020 06:17:16

%S 1,3,1,5,6,1,7,15,10,1,9,28,35,15,1,11,45,84,70,21,1,13,66,165,210,

%T 126,28,1,15,91,286,495,462,210,36,1,17,120,455,1001,1287,924,330,45,

%U 1,19,153,680,1820,3003,3003,1716,495,55,1,21,190,969,3060,6188

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A172431; see the Formula section.

%C Row sums: -1+odd-indexed Fibonacci numbers

%C Alternating row sums: 1,2,0,1,2,0,1,2,0,...

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

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

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

%F where u(1,x)=1, v(1,x)=1.

%e From _Paul Weisenhorn_, May 17 2020 : (Start)

%e First five rows of v(n.x):

%e   1

%e   3   1

%e   5   6   1

%e   7  15  10   1

%e   9  28  35  15  1

%e First three polynomials v(n,x): 1, 3 + x, 5 + 6x + x^2. (End)

%e From _Paul Weisenhorn_, May 14 2020: (Start)

%e First five rows of u(n,x):

%e   1

%e   1  2

%e   1  4   3

%e   1  6  10   4

%e   1  8  21  20  5

%e First three polynomials u(n,x): 1, 1 + 2x, 1 + 4x + 3x^2. (End)

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

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

%t v[n_, 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[%]    (* A172431 *)

%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[%]    (* A210551 *)

%Y Cf. A172431, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 22 2012