A274523 Number of integer partitions of n whose Durfee square has sides of even size.

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

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}]
(* A program translated from Maple: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
     If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]];
T[n_, k_] := T[n, k] = Sum[b[m, k]*b[n - k^2 - m, k], {m, 0, n - k^2}];
a[n_] := Sum[If[EvenQ[k], T[n, k], 0], {k, 0, Floor[Sqrt[n]]}];
a /@ Range[0, 70] (* Jean-François Alcover, May 31 2021, after Alois P. Heinz *)

Crossrefs

Cf. A115994.

Sequence in context: A215725 A022907 A006918 * A165189 A358055 A011842

Adjacent sequences: A274520 A274521 A274522 * A274524 A274525 A274526

Keyword

nonn

Author

Geoffrey Critzer, Jun 26 2016

Status

approved