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A119620 Number of partitions of floor(3n/2) into n parts each from {1,2,...,n}. 1
1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 11, 11, 15, 15, 22, 22, 30, 30, 42, 42, 56, 56, 77, 77, 101, 101, 135, 135, 176, 176, 231, 231, 297, 297, 385, 385, 490, 490, 627, 627, 792, 792, 1002, 1002, 1255, 1255, 1575, 1575, 1958, 1958, 2436, 2436, 3010, 3010, 3718, 3718 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The bisection {1,1,2,3,5,7,11,15,22,...} agrees with the initial terms of A008641, Number of partitions of n into at most 12 parts and also A008635, Molien series for A_12.

a(2n+1)=a(2n) for all n>0. If the partition {...,1} is a member of a(2n) then the partition {...,1,1} is a member of a(2n+1). - Robert G. Wilson v, Jun 09 2006

Number of partitions of n where all parts (except for possibly the first part) are even; see example. [Joerg Arndt, Apr 22 2013]

For n >= 2, a(n) = number of partitions p of n such that floor(n/2) is a part of p.  For n >= 1, a(n) = number of partitions p of n such that ceiling(n/2) is a part of p. - Clark Kimberling, Feb 28 2014

LINKS

Table of n, a(n) for n=0..57.

FORMULA

a(n) = A000041(floor(n/2)). - Vladeta Jovovic, Jun 10 2006

G.f.: ( Sum_{n>=0, x^(2*n) / prod(k=1..n, 1-x^k ) ) / (1 - x). - Michael Somos, Mar 01 2014

EXAMPLE

For n=8, Floor[3n/2] is 12 and there are five partitions of 12 into 8 parts each in the range 1-8 inclusive, namely: {5,1,1,1,1,1,1,1}, {4,2,1,1,1,1,1,1}, {3,3,1,1,1,1,1,1}, {3,2,2,1,1,1,1,1} and {2,2,2,2,1,1,1,1}. Thus a(8)=5.

From Joerg Arndt, Apr 22 2013: (Start)

a(8) = a(9) = 5, counting the following partitions where all parts (except for possibly the first part) are even:

01:  [ 2 2 2 2 ]

02:  [ 4 2 2 ]

03:  [ 4 4 ]

04:  [ 6 2 ]

05:  [ 8 ]

and

01:  [ 3 2 2 2 ]

02:  [ 5 2 2 ]

03:  [ 5 4 ]

04:  [ 7 2 ]

05:  [ 9 ]

(End)

G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + ...

MAPLE

# Using the function EULER from Transforms (see link at the bottom of the page).

[1, op(EULER([1, 0, seq(irem(n, 2), n=2..55)]))]; # Peter Luschny, Aug 19 2020

MATHEMATICA

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := f[n] = Length@ Select[ Partitions[ Floor[3n/2], n], Length@# == n &]; Table[ If[n > 1, f[2Floor[n/2]], f[n]], {n, 57}] (* Robert G. Wilson v, Jun 09 2006 *)

Table[ PartitionsP[ Floor[n/2]], {n, 57}] (* Robert G. Wilson v, Jun 09 2006 *)

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Ceiling[n/2]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)

a[ n_] := SeriesCoefficient[ (1 + x) / QPochhammer[x^2], {x, 0, n}]; (* Michael Somos, Mar 01 2014 *)

PROG

a(n)=numbpart(n\2); \\ Joerg Arndt, Apr 22 2013

CROSSREFS

Cf. A000041, A008641, A008635.

Sequence in context: A109763 A321523 A226748 * A240870 A265771 A239513

Adjacent sequences:  A119617 A119618 A119619 * A119621 A119622 A119623

KEYWORD

nonn

AUTHOR

John W. Layman, Jun 07 2006

EXTENSIONS

More terms from Robert G. Wilson v, Jun 09 2006

Added a(0)=1. - Michael Somos, Mar 01 2014

STATUS

approved

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Last modified September 21 05:55 EDT 2020. Contains 337267 sequences. (Running on oeis4.)