A209171 Triangle of coefficients of polynomials v(n,x) jointly generated with A209170; see the Formula section.

1, 3, 2, 6, 8, 3, 12, 25, 19, 5, 24, 68, 77, 40, 8, 48, 172, 259, 201, 80, 13, 96, 416, 782, 806, 478, 154, 21, 192, 976, 2200, 2825, 2222, 1067, 289, 34, 384, 2240, 5888, 9048, 8857, 5640, 2277, 532, 55, 768, 5056, 15184, 27160, 31787, 25184, 13483

Offset

1,2

Comments

Column 1: Fibonacci numbers (A000045).

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

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

Subtriangle of (1, 2, -3/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2012

Links

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

Formula

u(n,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)+1,

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 3, T(2,1) = 2. - Philippe Deléham, Mar 10 2012

Sum_{k=0..n} T(n,k)*x^k = A000012(n), A003945(n-1), A007483(n-1) for x = -1, 0, 1 respectively. - Philippe Deléham, Mar 10 2012

G.f.: (-1-x-x*y)*x*y/(-1+2*x+x*y+x^2*y^2+x^2*y). - R. J. Mathar, Aug 12 2015

Example

First five rows:
   1;
   3,  2;
   6,  8,  3;
  12, 25, 19,  5;
  24, 68, 77, 40,  8;
First three polynomials v(n,x):
  1
  3 + 2x
  6 + 8x + 3x^2.
From Philippe Deléham, Mar 10 2012: (Start)
Triangle (1, 2, -3/2, 1/2, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins (0 <= k <= n):
   1;
   1,   0;
   3,   2,   0;
   6,   8,   3,   0;
  12,  25,  19,   5,   0;
  24,  68,  77,  40,   8,   0;
  48, 172, 259, 201,  80,  13,   0;
  96, 416, 782, 806, 478, 154,  21,   0; (End)

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_] := (x + 1)*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[%]    (* A209170 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209171 *)

Crossrefs

Cf. A209170, A208510.

Sequence in context: A127717 A210236 A193998 * A368150 A348686 A160855

Adjacent sequences: A209168 A209169 A209170 * A209172 A209173 A209174

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 08 2012

Status

approved