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A210790 Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.

%I A210790 #17 Jan 27 2020 01:33:38
%S A210790 1,1,2,1,2,3,1,4,5,5,1,4,10,10,8,1,6,14,24,20,13,1,6,21,38,52,38,21,1,
%T A210790 8,27,65,96,109,71,34,1,8,36,92,176,224,220,130,55,1,10,44,136,280,
%U A210790 446,500,434,235,89,1,10,55,180,440,772,1066,1074,839,420,144,1
%N A210790 Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.
%C A210790 Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
%C A210790 Column 2: 2,2,4,4,6,6,8,8,...
%C A210790 Row sums: A105476.
%C A210790 Alternating row sums: signed Fibonacci numbers.
%C A210790 For a discussion and guide to related arrays, see A208510.
%C A210790 Subtriangle of the triangle given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 28 2012
%F A210790 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A210790 v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
%F A210790 where u(1,x)=1, v(1,x)=1.
%F A210790 From _Philippe Deléham_, Mar 28 2012: (Start)
%F A210790 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A210790 G.f.: (1+x-y*x-y^2*x^2)/(1-y*x-x^2-y*x^2-y^2*x^2).
%F A210790 T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A210790 First five rows:
%e A210790   1;
%e A210790   1,  2;
%e A210790   1,  2,  3;
%e A210790   1,  4,  5,  5;
%e A210790   1,  4, 10, 10,  8;
%e A210790 First three polynomials v(n,x):
%e A210790   1
%e A210790   1 + 2x
%e A210790   1 + 2x + 3x^2.
%e A210790 From _Philippe Deléham_, Mar 28 2012: (Start)
%e A210790 (1, 0, -1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins:
%e A210790   1;
%e A210790   1,  0;
%e A210790   1,  2,  0;
%e A210790   1,  2,  3,  0;
%e A210790   1,  4,  5,  5,  0;
%e A210790   1,  4, 10, 10,  8,  0; (End)
%t A210790 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210790 u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
%t A210790 d[x_] := h + x; e[x_] := p + x;
%t A210790 v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
%t A210790 j = 0; c = 0; h = 2; p = -1; f = 0;
%t A210790 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210790 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210790 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210790 TableForm[cu]
%t A210790 Flatten[%]    (* A210789 *)
%t A210790 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210790 TableForm[cv]
%t A210790 Flatten[%]    (* A210790 *)
%t A210790 Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A006138 *)
%t A210790 Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A105476 *)
%t A210790 Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)
%t A210790 Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)
%Y A210790 Cf. A210789, A208510.
%K A210790 nonn,tabl
%O A210790 1,3
%A A210790 _Clark Kimberling_, Mar 26 2012