login
Triangle of coefficients of polynomials v(n,x) jointly generated with A210801; see the Formula section.
3

%I #5 Mar 30 2012 18:58:17

%S 1,2,2,5,5,3,8,16,11,5,17,34,40,22,8,26,82,107,93,43,13,53,163,287,

%T 287,201,81,21,80,352,674,862,709,419,150,34,161,676,1592,2272,2326,

%U 1641,845,273,55,242,1378,3482,5878,6797,5863,3638,1666,491,89,485

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210801; see the Formula section.

%C Row n ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Row sums: A003462

%C Alternating row sums: A077898

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

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

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

%e 5....5....3

%e 8....16...11...5

%e 17...34...40...22...8

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

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

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

%t d[x_] := h + x; e[x_] := p + x;

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

%t j = 1; c = 1; h = 2; p = -1; f = 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[%] (* A210801 *)

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A210802 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A003462 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A003462 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077898 *)

%Y Cf. A210801, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 27 2012