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%I #13 Jan 26 2020 20:55:01
%S 1,2,1,3,4,2,6,10,8,3,9,24,27,16,5,18,51,74,62,30,8,27,108,189,200,
%T 136,56,13,54,216,450,574,488,282,102,21,81,432,1026,1536,1571,1128,
%U 569,184,34,162,837,2268,3864,4598,3967,2486,1118,328,55,243,1620
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210794; see the Formula section.
%C Row n starts with A038754(n) and ends with F(n), where F=A000045 (Fibonacci numbers).
%C Row sums: A000244 (powers of 3).
%C Alternating row sums: A000012 (1,1,1,1,1,1,1,1,1,1,1,...).
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle given by (1, 1, -1, -1, 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 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 29 2012: (Start)
%F As DELTA(triangle T(n,k) with 0 <= k <= n:
%F G.f.: (1 + x - y*x^2 - 2*y*x^2 - y^2*x^2)/(1 - y*x - 3*x^2 - 2*y*x^2 - y^2*x^2).
%F T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k <= n.
%e First five rows:
%e 1;
%e 2, 1;
%e 3, 4, 2;
%e 6, 10, 8, 3;
%e 9, 24, 27, 16, 5;
%e First three polynomials u(n,x):
%e 1
%e 2 + x
%e 3 + 4x + 2x^2.
%e From _Philippe Deléham_, Mar 29 2012: (Start)
%e (1, 1, -1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 2, 1, 0;
%e 3, 4, 2, 0;
%e 6, 10, 8, 3, 0;
%e 9, 24, 27, 16, 5, 0;
%e 18, 51, 74, 62, 30, 8, 0; (End)
%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 = 1; c = 0; h = 2; p = -1; 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[%] (* A210793 *)
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A210794 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
%Y Cf. A210794, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 26 2012
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