A209144 - OEIS

%I #16 Jan 22 2020 02:42:23

%S 1,3,6,1,12,5,24,16,1,48,44,7,96,112,30,1,192,272,104,9,384,640,320,

%T 48,1,768,1472,912,200,11,1536,3328,2464,720,70,1,3072,7424,6400,2352,

%U 340,13,6144,16384,16128,7168,1400,96,1,12288,35840,39680,20736

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.

%C Alternating row sums: 1,3,5,7,9,11,13,15,17,...

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

%C Subtriangle of the triangle given by (3,-1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 07 2012

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

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

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

%F From _Philippe Deléham_, Mar 07 2012: (Start)

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

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

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

%F Sum_{k=0..n} T(n,k)*x^k = A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. (End)

%e First five rows:

%e    1;

%e    3;

%e    6,  1;

%e   12,  5;

%e   24, 16, 1;

%e First three polynomials v(n,x): 1, 3, 6 + x.

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

%e     1;

%e     3,   0;

%e     6,   1,   0;

%e    12,   5,   0, 0;

%e    24,  16,   1, 0, 0;

%e    48,  44,   7, 0, 0, 0;

%e    96, 112,  30, 1, 0, 0, 0;

%e   192, 272, 104, 9, 0, 0, 0, 0;

%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_] := u[n - 1, x] + v[n - 1, x] + 1;

%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[%]    (* A209143 *)

%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[%]    (* A209144 *)

%Y Cf. A209143, A208510.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Mar 06 2012