A210042 Triangle of coefficients of polynomials u(n,x) jointly generated with A124927; see the Formula section.

1, 3, 5, 2, 7, 6, 2, 9, 12, 8, 2, 11, 20, 20, 10, 2, 13, 30, 40, 30, 12, 2, 15, 42, 70, 70, 42, 14, 2, 17, 56, 112, 140, 112, 56, 16, 2, 19, 72, 168, 252, 252, 168, 72, 18, 2, 21, 90, 240, 420, 504, 420, 240, 90, 20, 2, 23, 110, 330, 660, 924, 924, 660, 330, 110

Offset

1,2

Comments

Row sums: A000225

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

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

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

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

Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

Links

Table of n, a(n) for n=1..65.

Formula

First five rows:

1;

3,

5, 2;

7, 6, 2;

9, 12, 8, 2;

First three polynomials u(n,x): 1, 3, 5 + 2x.

Also, counting the top row as row 0, row n for n > 0 is as follows: 2n+1, 2C(n,2), 2C(n,3), ..., 2C(n,n).

From Philippe Deléham, Mar 25 2012: (Start)

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

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

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

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

Example

From Philippe Deléham, Mar 25 2012: (Start)
(1, 2, -2, 1, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:
   1;
   1,  0;
   3,  0,  0;
   5,  2,  0,  0;
   7,  6,  2,  0,  0;
   9, 12,  8,  2,  0,  0;
  11, 20, 20, 10,  2,  0,  0;
  13, 30, 40, 30, 12,  2,  0,  0; (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*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[%]    (* A210042 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A124927 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A010701 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 signed *)

Crossrefs

Cf. A124927, A208510.

Sequence in context: A097465 A340272 A120683 * A338872 A274421 A079313

Adjacent sequences: A210039 A210040 A210041 * A210043 A210044 A210045

Keyword

nonn,tabf

Author

Clark Kimberling, Mar 17 2012

Status

approved