A209149 Triangle of coefficients of polynomials v(n,x) jointly generated with A209146; see the Formula section.

1, 3, 1, 6, 5, 1, 12, 16, 7, 1, 24, 44, 30, 9, 1, 48, 112, 104, 48, 11, 1, 96, 272, 320, 200, 70, 13, 1, 192, 640, 912, 720, 340, 96, 15, 1, 384, 1472, 2464, 2352, 1400, 532, 126, 17, 1, 768, 3328, 6400, 7168, 5152, 2464, 784, 160, 19, 1, 1536, 7424

Offset: 1

Author

Clark Kimberling, Mar 07 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums: 1,2,2,2,2,2,2,2,2,2,2,2,2,...

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

As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012

A skew triangle of A209144. - Philippe Deléham, Mar 08 2012

Riordan array ( (1 + x)/(1 - 2*x), x/(1 - 2*x) ). Cf. A118800. Matrix inverse is a signed version of A112626. - Peter Bala, Jul 17 2013

Examples

			First five rows:
   1;
   3,  1;
   6,  5,  1;
  12, 16,  7, 1;
  24, 44, 30, 9, 1;
First three polynomials v(n,x): 1, 3 + x, 6 + 5x + x^2.
v(1,x) = 1
v(2,x) = 3 + x
v(3,x) = (3 + x)*(2 + x)
v(4,x) = (3 + x)*(2 + x)^2
v(5,x) = (3 + x)*(2 + x)^3
v(n,x) = (3 + x)*(2 + x)^(n-2)for n > 1. - _Philippe Deléham_, Mar 08 2012
		

Crossrefs

Cf. A209144, A209146, A209148, A208510.

Cf. A084938, A112626.

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] + (x + 1)*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[%]    (* A209148 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209149 *)

Formula

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

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

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

As DELTA-triangle:

T(n,k) = 2*T(n-1,k) + T(n-1,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 08 2012

As DELTA-triangle: G.f. is (1+x)/(1-2*x-yx). - Philippe Deléham, Mar 08 2012