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

1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 6, 17, 15, 16, 1, 8, 23, 49, 31, 32, 1, 9, 39, 72, 129, 63, 64, 1, 11, 48, 150, 201, 321, 127, 128, 1, 12, 70, 198, 501, 522, 769, 255, 256, 1, 14, 82, 338, 699, 1524, 1291, 1793, 511, 512, 1, 15, 110, 420, 1375, 2223, 4339, 3084, 4097, 1023, 1024

Offset: 1

Author

Clark Kimberling, Mar 08 2012

Keywords

• nonn
• tabl

Comments

For n > 1, n-th alternating row sum = (-1)^(n-1)*F(2*n-3), where F=A000045 (Fibonacci numbers).

Coefficient of x^(n-1) in u(n,x): 2^(n-1).

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

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

Examples

			First five rows:
  1;
  1,  2;
  1,  3,  4;
  1,  5,  7,  8;
  1,  6, 17, 15, 16;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 3x + 4x^2.
From _Philippe Deléham_, Mar 11 2012: (Start)
(1, 0, -1/2, -1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   3,   4,   0;
  1,   5,   7,   8,   0;
  1,   6,  17,  15,  16,   0;
  1,   8,  23,  49,  31,  32,   0;
  1,   9,  39,  72, 129,  63,  64,   0;
  1,  11,  48, 150, 201, 321, 127, 128,   0; (End)
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A209172, A208510.

Programs

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 *)
CoefficientList[CoefficientList[Series[(1 + x - 3*y*x - y*x^2 + 2*y^2*x^2)/(1 - 3*y*x - (1 - 2 y^2)*x^2), {x,0,10}, {y,0,10}], x],y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

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), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(2,0) = 1, T(2,1) = 2, T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.

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