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A210560 Triangle of coefficients of polynomials v(n,x) jointly generated with A210559; see the Formula section. 4

%I

%S 1,3,1,5,4,2,7,9,9,3,9,16,23,16,5,11,25,46,48,30,8,13,36,80,110,101,

%T 54,13,15,49,127,215,257,203,97,21,17,64,189,378,552,570,401,172,34,

%U 19,81,268,616,1057,1337,1228,776,303,55,21,100,366,948,1862,2772

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210559; see the Formula section.

%C Column 1: odd positive integers (A005408)

%C Column 2: squares (A000290)

%C Row n ends with F(n), where F=A000045 (Fibonacci numbers)

%C Row sums: A005409

%C Alternating row sums: 1,2,3,4,5,6,7,8,...(A000027)

%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)=(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...4...2

%e 7...9...9...3

%e 9...16...23...16...5

%e First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 2x^2.

%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_] := (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[%] (* A210559 *)

%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[%] (* A210560 *)

%Y Cf. A210559, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 22 2012

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Last modified November 12 14:43 EST 2019. Contains 329058 sequences. (Running on oeis4.)