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

Author

Clark Kimberling, Mar 13 2012

Keywords

• nonn
• tabl

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

Examples

			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.
		

Crossrefs

Cf. A000045, A000079, A209745, A208510.

Programs

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 *)

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