A209416 Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section.

1, 1, 2, 1, 3, 3, 1, 5, 7, 4, 1, 6, 15, 14, 5, 1, 8, 23, 36, 25, 6, 1, 9, 36, 69, 76, 41, 7, 1, 11, 48, 123, 176, 147, 63, 8, 1, 12, 66, 192, 355, 400, 266, 92, 9, 1, 14, 82, 292, 635, 910, 834, 456, 129, 10, 1, 15, 105, 410, 1065, 1833, 2131, 1626, 747, 175, 11

Offset

1,3

Comments

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

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

Subtriangle of the triangle given by (1, 0, -1/2, -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, 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) + x*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-2*y*x-y*x^2+y^2*x^2)/((1-2*y*x-x^2-y*x^2+y^2*x^2).

T(n,k) = 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, 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,  2;
  1,  3,  3;
  1,  5,  7,  4;
  1,  6, 15, 14,  5;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 3x + 3x^2.
From Philippe Deléham, Apr 01 2012: (Start)
(1, 0, -1/2, -1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  3,  3,  0;
  1,  5,  7,  4,  0;
  1,  6, 15, 14,  5,  0;
  1,  8, 23, 36, 25,  6,  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] + 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[%]    (* A209415 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209416 *)
CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

Crossrefs

Cf. A209416, A208510.

Sequence in context: A100578 A061315 A144265 * A122075 A185675 A153341

Adjacent sequences: A209413 A209414 A209415 * A209417 A209418 A209419

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 09 2012

Status

approved