A119620 Number of partitions of floor(3n/2) into n parts each from {1,2,...,n}.

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

Offset: 0

Author

John W. Layman, Jun 07 2006

Keywords

• nonn
• easy

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

From Gus Wiseman, Oct 28 2021: (Start)

If we insert zeros every three terms, this counts partitions of n such that n = floor(3*k/2), where k is the number of parts. This counts by sum rather than length. These partitions are ranked by A347452.

Also the number of integer partitions of n with alternating product 1, where the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). These are the conjugates of the partitions (ranked by A336119) described in Arndt's comment above. For example, the a(2) = 1 through a(10) = 7 partitions are:

11 111 22 221 33 331 44 441 55

1111 11111 2211 22111 2222 22221 3322

111111 1111111 3311 33111 4411

221111 2211111 222211

11111111 111111111 331111

22111111

1111111111

These partitions are ranked by A028982. The odd-length case is A035363 (shifted), which is also the version for sum instead of product. The multiplicative version (factorizations) is A347438.

(End)

Examples

			For n=8, floor(3*n/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 + ...
		

Crossrefs

Cf. A008641, A008635.

Both bisections are A000041.

An adjoint version is A108711.

A027187 counts partitions of even length.

A027193 counts partitions of odd length.

A325534 counts separable partitions.

A325535 counts inseparable partitions.

Cf. A000070, A001700, A028983, A086543, A182616, A236559, A236914, A344654, A347444, A347452.

Programs

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 *)

PARI

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

Formula

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

G.f.: (Sum_{n>=0} x^(4*n) / Product_{k=1..n} (1-x^(2*k))) / (1 - x). - Michael Somos, Mar 01 2014 [corrected by Jason Yuen, Jan 24 2025]

Extensions

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

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