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

%I A210792 #13 Jan 26 2020 20:55:18
%S A210792 1,1,2,1,5,3,1,10,11,5,1,19,28,25,8,1,36,62,81,50,13,1,69,129,218,193,
%T A210792 98,21,1,134,261,533,597,442,185,34,1,263,522,1235,1631,1559,952,343,
%U A210792 55,1,520,1040,2773,4129,4763,3758,1985,625,89,1,1033,2071
%N A210792 Triangle of coefficients of polynomials v(n,x) jointly generated with A210791; see the Formula section.
%C A210792 Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
%C A210792 Column 2: A052944.
%C A210792 Row sums: A000244 (powers of 3).
%C A210792 Alternating row sums: A001333.
%C A210792 For a discussion and guide to related arrays, see A208510.
%C A210792 Subtriangle of the triangle given by (1, 0, 1/2, 3/2, 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 29 2012
%F A210792 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A210792 v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),
%F A210792 where u(1,x)=1, v(1,x)=1.
%F A210792 From _Philippe Deléham_, Mar 29 2012: (Start)
%F A210792 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A210792 G.f.: (1 - 2*x - y*x + 3*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).
%F A210792 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) = 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 A210792 First five rows:
%e A210792   1;
%e A210792   1,  2;
%e A210792   1,  5,  3;
%e A210792   1, 10, 11,  5;
%e A210792   1, 19, 28, 25,  8;
%e A210792 First three polynomials v(n,x):
%e A210792   1
%e A210792   1 + 2x
%e A210792   1 + 5x + 3x^2
%e A210792 From _Philippe Deléham_, Mar 29 2012: (Start)
%e A210792 (1, 0, 1/2, 3/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins:
%e A210792   1;
%e A210792   1,   0;
%e A210792   1,   2,   0;
%e A210792   1,   5,   3,   0;
%e A210792   1,  10,  11,   5,   0;
%e A210792   1,  19,  28,  25,   8,   0;
%e A210792   1,  36,  62,  81,  50,  13,   0;
%e A210792   1,  69, 129, 218, 193,  98,  21,   0; (End)
%t A210792 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210792 u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
%t A210792 d[x_] := h + x; e[x_] := p + x;
%t A210792 v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
%t A210792 j = 0; c = 0; h = -1; p = 2; f = 0;
%t A210792 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210792 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210792 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210792 TableForm[cu]
%t A210792 Flatten[%]    (* A210791 *)
%t A210792 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210792 TableForm[cv]
%t A210792 Flatten[%]    (* A210792 *)
%t A210792 Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A007051 *)
%t A210792 Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)
%t A210792 Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A001129 *)
%t A210792 Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A001333 *)
%Y A210792 Cf. A210791, A208510.
%K A210792 nonn,tabl
%O A210792 1,3
%A A210792 _Clark Kimberling_, Mar 26 2012