A209831 - OEIS

%I #14 Jan 26 2020 21:06:02

%S 1,1,3,1,5,8,1,8,20,21,1,10,41,71,55,1,13,65,176,235,144,1,15,99,338,

%T 684,744,377,1,18,135,590,1536,2490,2285,987,1,20,182,926,3031,6382,

%U 8651,6865,2584,1,23,230,1388,5359,14065,24875,29020,20284,6765

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209830; see the Formula section.

%C Each row begins with 1 and ends with an even-indexed Fibonacci number.

%C Alternating row sums: signed powers of 2.

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

%C Subtriangle of the triangle given by (1, 0, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 16 2012

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

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

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

%F As DELTA-triangle T(n,k) with 0 <= k <= n:

%F T(n,k) = 3*T(n-1,k-1) + 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(1,1) = T(2,2) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 16 2012

%F As DELTA-triangle with 0 <= k <= n: g.f.: (1 + x - 3*y*x - 2*y*x^2 + y^2*x^2)/(1 - 3*y*x - x^2 - 2*y*x^2 + y^2*x^2). - _Philippe Deléham_, Mar 16 2012

%e From _Philippe Deléham_, Mar 16 2012: (Start)

%e First five rows:

%e   1;

%e   1,  3;

%e   1,  5,  8;

%e   1,  8, 20, 21;

%e   1, 10, 41, 71, 55;

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

%e   1

%e   1 + 3x

%e   1 + 5x + 8x^2.

%e (1, 0, -1/3, -2/3, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, ...) begins:

%e   1;

%e   1,  0;

%e   1,  3,  0;

%e   1,  5,  8,  0;

%e   1,  8, 20, 21,  0;

%e   1, 10, 41, 71, 55,  0; (End)

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

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

%t v[n_, x_] := (x + 1)*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[%]    (* A209830 *)

%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[%]    (* A209831 *)

%Y Cf. A209830, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 13 2012