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A274523 Number of integer partitions of n whose Durfee square has sides of even size. 1
1, 0, 0, 0, 1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, 112, 141, 170, 209, 250, 305, 364, 444, 534, 655, 796, 984, 1208, 1504, 1860, 2322, 2882, 3597, 4460, 5546, 6852, 8471, 10406, 12773, 15584, 18984, 22994, 27794, 33422, 40099, 47882, 57046, 67676, 80111 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

A partition of n has a Durfee square of side s if s is the largest number such that the partition contains at least s parts with values >= s.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Aubrey Blecher, Arnold Knopfmacher and Augustine Munagi, Durfee square areas and associated partition identities, 2014.

David A. Corneth, Illustration of a(6)

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 45.

Ljuben R. Mutafchiev, On the size of the Durfee square of a random integer partition, Journal of Computational and Applied Mathematics, Volume 142, Issue 1, 1 May 2002, Pages 173-184.

Wikipedia, Durfee square.

FORMULA

G.f.: Sum_{k>=0} x^((2k)^2)/ Product_{i=1..2k} (1-x^i)^2.

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, May 21 2018

EXAMPLE

a(6)=5 because we have: 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1 all having a Durfee square of side s=2.

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1,

      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

    end:

T:= proc(n, k) option remember;

      add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2)

    end:

a:= n-> add(`if`(k::even, T(n, k), 0), k=0..floor(sqrt(n))):

seq(a(n), n=0..70);  # Alois P. Heinz, Jun 27 2016

MATHEMATICA

nn = 40; CoefficientList[ Series[Sum[z^((2 h)^2)/Product[(1 - z^i), {i, 1, 2 h}]^2, {h, 0, nn}], {z, 0, nn}], z]

(* or by brute force *)

Table[Count[Map[EvenQ, Map[DurfeeSquare, IntegerPartitions[n]]],

  True], {n, 0, 30}]

CROSSREFS

Cf. A115994.

Sequence in context: A215725 A022907 A006918 * A165189 A011842 A000094

Adjacent sequences:  A274520 A274521 A274522 * A274524 A274525 A274526

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Jun 26 2016

STATUS

approved

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Last modified July 16 06:24 EDT 2018. Contains 312654 sequences. (Running on oeis4.)