A210040 Array of coefficients of polynomials v(n,x) jointly generated with A210039; see the Formula section.

1, 2, 1, 3, 4, 4, 10, 1, 5, 20, 6, 6, 35, 21, 1, 7, 56, 56, 8, 8, 84, 126, 36, 1, 9, 120, 252, 120, 10, 10, 165, 462, 330, 55, 1, 11, 220, 792, 792, 220, 12, 12, 286, 1287, 1716, 715, 78, 1, 13, 364, 2002, 3432, 2002, 364, 14, 14, 455, 3003, 6435, 5005, 1365

Offset: 1

Author

Clark Kimberling, Mar 17 2012

Keywords

• nonn
• tabf

Comments

Every term is a binomial coefficient.

Row sums: A000225

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

Examples

			First eight rows:
1
2...1
3...4
4...10...1
5...20...6
6...35...21....1
7...56...56....8
8...84...126...36...1
First five polynomials v(n,x):
1
2 + x
3 + 4x.
4 + 10x + x^2
5 + 20x + 6x^2.
		

Crossrefs

Cf. A210039, 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] + 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[%]    (* A210039 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210040 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

Formula

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

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

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

Also: writing T(n,m) for the general term,

T(n,1)=n for n>=1;

T(n,k)=C(n+1,2k-1) for 1<=k<=floor[(n+2)/2].