A209746 Triangle of coefficients of polynomials v(n,x) jointly generated with A209745; see the Formula section.

1, 2, 2, 3, 7, 4, 5, 17, 20, 8, 8, 37, 65, 52, 16, 13, 75, 176, 210, 128, 32, 21, 146, 428, 679, 616, 304, 64, 34, 276, 971, 1921, 2312, 1696, 704, 128, 55, 511, 2097, 4970, 7449, 7240, 4464, 1600, 256, 89, 931, 4366, 12056, 21622, 26146, 21344

Offset

1,2

Comments

Row n begins with F(n+1) and ends with 2^(n-1), where F=A000045 (Fibonacci numbers).

Alternating row sums: 1,0,0,0,0,0,0,0,0,0,...

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

Riordan array ((1+x)/(1-x-x^2), (2x+x^2)/(1-x-x^2)). - Philippe Deléham, Mar 24 2012

Triangle given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012

Links

Table of n, a(n) for n=1..52.

Formula

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

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

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k >= 0. - Philippe Deléham, Mar 24 2012

Example

First five rows:
  1;
  2,  2;
  3,  7,  4;
  5, 17, 20,  8;
  8, 37, 65, 52, 16;
First three polynomials v(n,x):
  1
  2 + 2x
  3 + 7x + 4x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
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[%]    (* A209745 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209746 *)

Crossrefs

Cf. A000045, A000079, A209745, A208510.

Sequence in context: A222894 A210753 A208762 * A354950 A267822 A210598

Adjacent sequences: A209743 A209744 A209745 * A209747 A209748 A209749

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 13 2012

Status

approved