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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
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
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
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jun 26 2016
STATUS
approved