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

Author

Clark Kimberling, Mar 08 2012

Keywords

• nonn
• tabl

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

Examples

			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)
		

Crossrefs

Cf. A209170, A208510.

Programs

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 *)

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