A209144 Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.

1, 3, 6, 1, 12, 5, 24, 16, 1, 48, 44, 7, 96, 112, 30, 1, 192, 272, 104, 9, 384, 640, 320, 48, 1, 768, 1472, 912, 200, 11, 1536, 3328, 2464, 720, 70, 1, 3072, 7424, 6400, 2352, 340, 13, 6144, 16384, 16128, 7168, 1400, 96, 1, 12288, 35840, 39680, 20736

Offset: 1

Author

Clark Kimberling, Mar 06 2012

Keywords

• nonn
• tabf

Comments

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

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

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

Examples

			First five rows:
   1;
   3;
   6,  1;
  12,  5;
  24, 16, 1;
First three polynomials v(n,x): 1, 3, 6 + x.
(3,-1, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, ...) begins:
    1;
    3,   0;
    6,   1,   0;
   12,   5,   0, 0;
   24,  16,   1, 0, 0;
   48,  44,   7, 0, 0, 0;
   96, 112,  30, 1, 0, 0, 0;
  192, 272, 104, 9, 0, 0, 0, 0;
		

Crossrefs

Cf. A209143, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209143 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209144 *)

Formula

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

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

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

From Philippe Deléham, Mar 07 2012: (Start)

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

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.

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

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)