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A210791 Triangle of coefficients of polynomials u(n,x) jointly generated with A210792; see the Formula section.

%I A210791 #13 Jan 26 2020 20:55:36
%S A210791 1,1,1,1,2,2,1,3,7,3,1,4,17,14,5,1,5,36,42,30,8,1,6,72,104,111,58,13,
%T A210791 1,7,141,233,329,251,111,21,1,8,275,494,862,848,553,206,34,1,9,538,
%U A210791 1016,2097,2479,2112,1158,377,55,1,10,1058,2056,4870,6608,6875
%N A210791 Triangle of coefficients of polynomials u(n,x) jointly generated with A210792; see the Formula section.
%C A210791 Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
%C A210791 Column 2: 1,2,3,4,5,6,7,8,...
%C A210791 Row sums: A007051.
%C A210791 Alternating row sums: A000129.
%C A210791 For a discussion and guide to related arrays, see A208510.
%C A210791 Subtriangle of the triangle given by (1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 29 2012
%F A210791 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A210791 v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),
%F A210791 where u(1,x)=1, v(1,x)=1.
%F A210791 From _Philippe Deléham_, Mar 29 2012: (Start)
%F A210791 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A210791 G.f.: (1 - 2*x - y*x + 2*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).
%F A210791 T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A210791 First five rows:
%e A210791   1;
%e A210791   1,  1;
%e A210791   1,  2,  2;
%e A210791   1,  3,  7,  3;
%e A210791   1,  4, 17, 14,  5;
%e A210791 First three polynomials u(n,x):
%e A210791   1
%e A210791   1 + x
%e A210791   1 + 2x + 2x^2.
%e A210791 From _Philippe Deléham_, Mar 29 2012: (Start)
%e A210791 (1, 0, 0, 2, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
%e A210791   1;
%e A210791   1,   0;
%e A210791   1,   1,   0;
%e A210791   1,   2,   2,   0;
%e A210791   1,   3,   7,   3,   0;
%e A210791   1,   4,  17,  14,   5,   0;
%e A210791   1,   5,  36,  42,  30,   8,   0;
%e A210791   1,   6,  72, 104, 111,  58,  13,   0; (End)
%t A210791 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210791 u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
%t A210791 d[x_] := h + x; e[x_] := p + x;
%t A210791 v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
%t A210791 j = 0; c = 0; h = -1; p = 2; f = 0;
%t A210791 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210791 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210791 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210791 TableForm[cu]
%t A210791 Flatten[%]    (* A210791 *)
%t A210791 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210791 TableForm[cv]
%t A210791 Flatten[%]    (* A210792 *)
%t A210791 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A007051 *)
%t A210791 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
%t A210791 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A001129 *)
%t A210791 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A001333 *)
%Y A210791 Cf. A210792, A208510.
%K A210791 nonn,tabl
%O A210791 1,5
%A A210791 _Clark Kimberling_, Mar 26 2012