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A210039 Array of coefficients of polynomials u(n,x) jointly generated with A210040; see the Formula section. 3
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 (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..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: A110119 A322310 A152202 * A026250 A210033 A209144

Adjacent sequences:  A210036 A210037 A210038 * A210040 A210041 A210042

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Mar 17 2012

STATUS

approved

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Last modified October 17 19:18 EDT 2019. Contains 328127 sequences. (Running on oeis4.)