A209172 - OEIS

%I #15 Jan 24 2020 03:29:52

%S 1,1,1,1,3,1,1,4,7,1,1,6,11,15,1,1,7,23,26,31,1,1,9,30,72,57,63,1,1,

%T 10,48,102,201,120,127,1,1,12,58,198,303,522,247,255,1,1,13,82,256,

%U 699,825,1291,502,511,1,1,15,95,420,955,2223,2116,3084,1013,1023,1

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209413; see the Formula section.

%C For n > 1, n-th alternating row sum = ((-1)^n)*F(2n-4), where F=A000045 (Fibonacci numbers).  For a discussion and guide to related arrays, see A208510.

%C Subtriangle of the triangle given by (1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 11 2012

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

%F v(n,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 11 2012: (Start)

%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-2) with T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.

%F G.f.: (1+x-3*y*x-2*y*x^2+2*y^2*x^2)/(1-3*y*x-x^2+2*y^2*x^2). (End)

%e First five rows:

%e   1;

%e   1,  1;

%e   1,  3,  1;

%e   1,  4,  7,  1;

%e   1,  6, 11, 15,  1;

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

%e   1

%e   1 + x

%e   1 + 3x + x^2.

%e From _Philippe Deléham_, Mar 11 2012: (Start)

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

%e   1;

%e   1,  0;

%e   1,  1,  0;

%e   1,  3,  1,  0;

%e   1,  4,  7,  1,  0;

%e   1,  6, 11, 15,  1,  0;

%e   1,  7, 23, 26, 31,  1,  0;

%e   1,  9, 30, 72, 57, 63,  1,  0; (End)

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

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

%t v[n_, 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[%]    (* A209172 *)

%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[%]    (* A209413 *)

%Y Cf. A209413, A208510.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Mar 08 2012