A209418 Triangle of coefficients of polynomials v(n,x) jointly generated with A209417; see the Formula section.

1, 1, 3, 1, 4, 7, 1, 7, 13, 15, 1, 8, 30, 38, 31, 1, 11, 42, 104, 103, 63, 1, 12, 69, 178, 321, 264, 127, 1, 15, 87, 331, 657, 921, 649, 255, 1, 16, 124, 484, 1354, 2200, 2512, 1546, 511, 1, 19, 148, 760, 2266, 4978, 6856, 6598, 3595, 1023, 1, 20, 195, 1020, 3870, 9384, 16938, 20226, 16827, 8204, 2047

Offset

1,3

Comments

Alternating row sums: signed powers of 2.

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

Subtriangle of the triangle given by (1, 0, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012

Links

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

Formula

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

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

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

From Philippe Deléham, Apr 01 2012: (Start)

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

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

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

Example

First five rows:
  1;
  1,  3;
  1,  4,  7;
  1,  7, 13, 15;
  1,  8, 30, 38, 31;
First three polynomials v(n,x):
  1
  1 + 3x
  1 + 4x + 7x^2.
From Philippe Deléham, Apr 01 2012: (Start)
(1, 0, -2/3, -1/3, 0, 0, 0, ...) DELTA (0, 3, -2/3, 2/3, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  3,  0;
  1,  4,  7,  0;
  1,  7, 13, 15,  0;
  1,  8, 30, 38, 31,  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_] := (x + 1)*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[%]    (* A209417 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209418 *)
CoefficientList[CoefficientList[Series[(1 + x)/(1 - 3*y*x - x^2 - y*x^2 + 2*y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

Crossrefs

Cf. A209417, A208510.

Sequence in context: A328463 A185722 A287376 * A193969 A169838 A249271

Adjacent sequences: A209415 A209416 A209417 * A209419 A209420 A209421

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 09 2012

Status

approved