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Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.
3

%I #16 Aug 12 2015 14:21:08

%S 1,1,1,1,3,2,1,8,10,3,1,22,37,21,5,1,63,125,100,45,8,1,185,409,410,

%T 260,88,13,1,550,1321,1562,1240,598,169,21,1,1644,4238,5706,5331,3258,

%U 1319,315,34,1,4925,13534,20284,21507,15651,8071,2776,578,55,1

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.

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

%C Column 1: 1,1,1,1,1,1,1,1,1,1,1,...

%C Column 2: A047849

%C Row sums: A003462

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

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

%C Essentially the same triangle as (1, 0, 3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jul 11 2012

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

%F v(n,x)=(x-1)*u(n-1,x)+(x+3)*v(n-1,x),

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

%F T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = T(3,0) = 1, T(3,1) = 3, T(3,2) = 2, T(n,k) = 0 if k<0 or if k >= n. - _Philippe Deléham_, Jul 11 2012

%F G.f.: (-1+3*x)*x*y/(-1+4*x-3*x^2-2*x^2*y+x*y+x^2*y^2). - _R. J. Mathar_, Aug 12 2015

%e First five rows:

%e 1

%e 1...1

%e 1...3....2

%e 1...8....10...3

%e 1...22...37...21...5

%e First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^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 = 0; c = 0; h = -1; p = 3; f = 0;

%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[%] (* A210803 *)

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

%t TableForm[cv]

%t Flatten[%] (* A210804 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A047849 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000302 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000007 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000007 *)

%Y Cf. A210804, A208510.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Mar 27 2012