A210189 Triangle of coefficients of polynomials u(n,x) jointly generated with A210190; see the Formula section.

1, 3, 6, 2, 10, 10, 15, 30, 4, 21, 70, 28, 28, 140, 112, 8, 36, 252, 336, 72, 45, 420, 840, 360, 16, 55, 660, 1848, 1320, 176, 66, 990, 3696, 3960, 1056, 32, 78, 1430, 6864, 10296, 4576, 416, 91, 2002, 12012, 24024, 16016, 2912, 64, 105, 2730, 20020

Offset: 1

Author

Clark Kimberling, Mar 18 2012

Keywords

• nonn
• tabf

Comments

Column 1: triangular numbers: A000217

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

Examples

			First five rows:
1
3
6....2
10...10
15...30...4
First three polynomials u(n,x): 1, 3, 6 + 2x.
		

Crossrefs

Cf. A210190, 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_] := 2 x*u[n - 1, 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[%]   (* A210189 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210190 *)

Formula

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

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

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