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Number of integer partitions of n whose Durfee square has sides of even size.
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%I #43 May 31 2021 05:36:03

%S 1,0,0,0,1,2,5,8,14,20,30,40,55,70,91,112,141,170,209,250,305,364,444,

%T 534,655,796,984,1208,1504,1860,2322,2882,3597,4460,5546,6852,8471,

%U 10406,12773,15584,18984,22994,27794,33422,40099,47882,57046,67676,80111

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

%C 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.

%H Alois P. Heinz, <a href="/A274523/b274523.txt">Table of n, a(n) for n = 0..10000</a>

%H Aubrey Blecher, Arnold Knopfmacher and Augustine Munagi, <a href="https://www.researchgate.net/publication/287067866_Durfee_square_areas_and_associated_partition_identities">Durfee square areas and associated partition identities</a>, 2014.

%H David A. Corneth, <a href="/A274523/a274523.png">Illustration of a(6)</a>

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 45.

%H Ljuben R. Mutafchiev, <a href="http://dx.doi.org/10.1016/S0377-0427(01)00467-8">On the size of the Durfee square of a random integer partition</a>, Journal of Computational and Applied Mathematics, Volume 142, Issue 1, 1 May 2002, Pages 173-184.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Durfee_square">Durfee square</a>.

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

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, May 21 2018

%e 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.

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

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

%p end:

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

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

%p end:

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

%p seq(a(n), n=0..70); # _Alois P. Heinz_, Jun 27 2016

%t 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]

%t (* or by brute force *)

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

%t True], {n, 0, 30}]

%t (* A program translated from Maple: *)

%t b[n_, i_] := b[n, i] = If[n == 0, 1,

%t If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]];

%t T[n_, k_] := T[n, k] = Sum[b[m, k]*b[n - k^2 - m, k], {m, 0, n - k^2}];

%t a[n_] := Sum[If[EvenQ[k], T[n, k], 0], {k, 0, Floor[Sqrt[n]]}];

%t a /@ Range[0, 70] (* _Jean-François Alcover_, May 31 2021, after _Alois P. Heinz_ *)

%Y Cf. A115994.

%K nonn

%O 0,6

%A _Geoffrey Critzer_, Jun 26 2016