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Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section.
3

%I #14 Jan 24 2020 03:27:27

%S 1,2,1,2,4,3,2,6,12,7,2,8,22,32,17,2,10,34,70,86,41,2,12,48,124,216,

%T 228,99,2,14,64,196,428,644,600,239,2,16,82,288,744,1408,1876,1568,

%U 577,2,18,102,402,1188,2664,4476,5364,4074,1393,2,20,124,540,1786

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section.

%C For a discussion and guide to related arrays, see A208510.

%C Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012

%C Row sums are powers of 3 (A000244). - _Philippe Deléham_, Mar 21 2012

%F u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),

%F v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),

%F where u(1,x)=1, v(1,x)=1.

%F From _Philippe Deléham_, Mar 21 2012: (Start)

%F As DELTA-triangle with 0 <= k <= n:

%F G.f.: (1-2*y*x+x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).

%F T(n,k) = T(n-1,k-1) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

%e First five rows:

%e 1;

%e 2, 1;

%e 2, 4, 3;

%e 2, 6, 12, 7;

%e 2, 8, 22, 32, 17;

%e First three polynomials u(n,x):

%e 1

%e 2 + x

%e 2 + 4x + 3x^2

%e From _Philippe Deléham_, Mar 21 2012: (Start)

%e (1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:

%e 1;

%e 1, 0;

%e 2, 1, 0;

%e 2, 4, 3, 0;

%e 2, 6, 12, 7, 0;

%e 2, 8, 22, 32, 17, 0; (End)

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];

%t v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];

%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[%] (* A209128 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A209129 *)

%Y Cf. A209129, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 05 2012