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A210803 Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.

%I A210803 #16 Aug 12 2015 14:21:08
%S A210803 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 A210803 260,88,13,1,550,1321,1562,1240,598,169,21,1,1644,4238,5706,5331,3258,
%U A210803 1319,315,34,1,4925,13534,20284,21507,15651,8071,2776,578,55,1
%N A210803 Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.
%C A210803 Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
%C A210803 Column 1:  1,1,1,1,1,1,1,1,1,1,1,...
%C A210803 Column 2:  A047849
%C A210803 Row sums:  A003462
%C A210803 Alternating row sums:  1,0,0,0,0,0,0,0,0,...
%C A210803 For a discussion and guide to related arrays, see A208510.
%C A210803 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 A210803 u(n,x)=u(n-1,x)+x*v(n-1,x)+1,
%F A210803 v(n,x)=(x-1)*u(n-1,x)+(x+3)*v(n-1,x),
%F A210803 where u(1,x)=1, v(1,x)=1.
%F A210803 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 A210803 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 A210803 First five rows:
%e A210803 1
%e A210803 1...1
%e A210803 1...3....2
%e A210803 1...8....10...3
%e A210803 1...22...37...21...5
%e A210803 First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^2.
%t A210803 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210803 u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
%t A210803 d[x_] := h + x; e[x_] := p + x;
%t A210803 v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
%t A210803 j = 0; c = 0; h = -1; p = 3; f = 0;
%t A210803 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210803 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210803 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210803 TableForm[cu]
%t A210803 Flatten[%]    (* A210803 *)
%t A210803 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210803 TableForm[cv]
%t A210803 Flatten[%]    (* A210804 *)
%t A210803 Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A047849 *)
%t A210803 Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000302 *)
%t A210803 Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)
%t A210803 Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)
%Y A210803 Cf. A210804, A208510.
%K A210803 nonn,tabl
%O A210803 1,5
%A A210803 _Clark Kimberling_, Mar 27 2012