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A210040 Array of coefficients of polynomials v(n,x) jointly generated with A210039; see the Formula section. 3
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 (list; graph; refs; listen; history; text; internal format)
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..61.

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].

EXAMPLE

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.

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. A210039, A208510.

Sequence in context: A033799 A325324 A241745 * A285329 A289023 A085985

Adjacent sequences:  A210037 A210038 A210039 * A210041 A210042 A210043

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Mar 17 2012

STATUS

approved

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)