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

1, 3, 6, 1, 10, 5, 15, 15, 1, 21, 35, 7, 28, 70, 28, 1, 36, 126, 84, 9, 45, 210, 210, 45, 1, 55, 330, 462, 165, 11, 66, 495, 924, 495, 66, 1, 78, 715, 1716, 1287, 286, 13, 91, 1001, 3003, 3003, 1001, 91, 1, 105, 1365, 5005, 6435, 3003, 455, 15, 120, 1820

Offset

1,2

Comments

Every term is a binomial coefficient.

Row sums: A000225

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

Links

Table of n, a(n) for n=1..58.

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 the general term as T(n,m),

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

Example

First eight rows:
1
3
6....1
10...5
15...15....1
21...35....7
28...70....28...1
36...126...84...9
First five polynomials u(n,x):
1
3
6 + x
10 + 5x
21 + 35x + 7x^2.

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 *)

Crossrefs

Cf. A034839, A210040, A208510.

Sequence in context: A322310 A359279 A152202 * A026250 A210033 A209144

Adjacent sequences: A210036 A210037 A210038 * A210040 A210041 A210042

Keyword

nonn,tabf

Author

Clark Kimberling, Mar 17 2012

Status

approved