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

1,2

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

Links

Table of n, a(n) for n=1..57.

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

Example

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

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 *)

Crossrefs

Cf. A209144, A209146, A209148, A208510.

Cf. A084938, A112626.

Sequence in context: A133545 A210214 A322427 * A343062 A210602 A210801

Adjacent sequences: A209146 A209147 A209148 * A209150 A209151 A209152

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 07 2012

Status

approved