A209172 Triangle of coefficients of polynomials u(n,x) jointly generated with A209413; see the Formula section.

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 11, 15, 1, 1, 7, 23, 26, 31, 1, 1, 9, 30, 72, 57, 63, 1, 1, 10, 48, 102, 201, 120, 127, 1, 1, 12, 58, 198, 303, 522, 247, 255, 1, 1, 13, 82, 256, 699, 825, 1291, 502, 511, 1, 1, 15, 95, 420, 955, 2223, 2116, 3084, 1013, 1023, 1

Offset

1,5

Comments

For n > 1, n-th alternating row sum = ((-1)^n)*F(2n-4), where F=A000045 (Fibonacci numbers). For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 11 2012

Links

Table of n, a(n) for n=1..66.

Formula

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

v(n,x) = u(n-1,x) + 2x*v(n-1,x),

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

From Philippe Deléham, Mar 11 2012: (Start)

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

T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2) with T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.

G.f.: (1+x-3*y*x-2*y*x^2+2*y^2*x^2)/(1-3*y*x-x^2+2*y^2*x^2). (End)

Example

First five rows:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  7,  1;
  1,  6, 11, 15,  1;
First three polynomials v(n,x):
  1
  1 + x
  1 + 3x + x^2.
From Philippe Deléham, Mar 11 2012: (Start)
(1, 0, 1, -2, 0, 0, 0,...) DELTA (0, 1, 0, 2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  3,  1,  0;
  1,  4,  7,  1,  0;
  1,  6, 11, 15,  1,  0;
  1,  7, 23, 26, 31,  1,  0;
  1,  9, 30, 72, 57, 63,  1,  0; (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*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[%]    (* A209172 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209413 *)

Crossrefs

Cf. A209413, A208510.

Sequence in context: A209415 A058879 A208344 * A263950 A160870 A345279

Adjacent sequences: A209169 A209170 A209171 * A209173 A209174 A209175

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 08 2012

Status

approved