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

Author

Clark Kimberling, Mar 17 2012

Keywords

• nonn
• tabf

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

Examples

			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)
		

Crossrefs

Cf. A124927, A208510.

Programs

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 *)

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