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Triangle of coefficients of polynomials u(n,x) jointly generated with A208750; see the Formula section.
3

%I #14 Jan 24 2020 03:30:30

%S 1,1,2,1,6,2,1,12,10,4,1,20,32,24,4,1,30,80,88,36,8,1,42,170,256,180,

%T 72,8,1,56,322,644,660,384,104,16,1,72,560,1456,1992,1568,704,192,16,

%U 1,90,912,3024,5256,5360,3392,1344,272,32,1,110,1410,5856,12552

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208750; see the Formula section.

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

%C Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 14 2012

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

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

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

%F As DELTA-triangle: g.f.: (1-x-2*y^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - _Philippe Deléham_, Mar 14 2012

%F Recurrence: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > =n. - _Philippe Deléham_, Mar 14 2012

%e First five rows:

%e 1;

%e 1, 2;

%e 1, 6, 2;

%e 1, 12, 10, 4;

%e 1, 20, 32, 24, 4;

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

%e 1

%e 1 + 2x

%e 1 + 6x + 2x^2

%e 1 + 12x + 10x^2 + 4x^3

%e 1 + 20x + 32x^2 + 24x^3 + 4x^4

%e From _Philippe Deléham_, Mar 14 2012: (Start)

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

%e 1;

%e 1, 0;

%e 1, 2, 0;

%e 1, 6, 2, 0;

%e 1, 12, 10, 4, 0;

%e 1, 20, 32, 24, 4, 0;

%e 1, 30, 80, 88, 36, 8, 0; (End)

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

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

%t v[n_, x_] := (x + 1)*u[n - 1, 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[%] (* A208749 *)

%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[%] (* A208750 *)

%Y Cf. A208750, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 02 2012