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Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; see the Formula section.
3

%I #14 Jan 24 2020 03:30:02

%S 1,1,2,1,6,2,1,12,12,2,1,20,40,18,2,1,30,100,86,24,2,1,42,210,294,150,

%T 30,2,1,56,392,812,656,232,36,2,1,72,672,1932,2268,1240,332,42,2,1,90,

%U 1080,4116,6624,5172,2100,450,48,2,1,110,1650,8052,17028,17996

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; 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, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 17 2012

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

%F v(n,x) = (x+1)*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 17 2012: (Start)

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

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

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

%e First five rows:

%e 1;

%e 1, 2;

%e 1, 6, 2;

%e 1, 12, 12, 2;

%e 1, 20, 40, 18, 2;

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

%e 1

%e 1 + 2x

%e 1 + 6x + 2x^2

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

%e 1 + 20x + 40x^2 + 18x^3 + 2x^4

%e From _Philippe Deléham_, Mar 17 2012: (Start)

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

%e 1;

%e 1, 0;

%e 1, 2, 0;

%e 1, 6, 2, 0;

%e 1, 12, 12, 2, 0;

%e 1, 20, 40, 18, 2, 0;

%e 1, 30, 100, 86, 24, 2, 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_] := u[n - 1, x] + (x + 1) 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[%] (* A208751 *)

%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[%] (* A208752 *)

%Y Cf. A208752, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 01 2012