A116503 Sum of the areas of the Durfee squares of all partitions of n.

1, 2, 3, 8, 13, 26, 39, 64, 98, 148, 216, 322, 455, 648, 904, 1258, 1711, 2336, 3128, 4198, 5548, 7330, 9569, 12496, 16146, 20836, 26674, 34098, 43273, 54846, 69072, 86848, 108627, 135612, 168527, 209066, 258271, 318482, 391321, 479946, 586709

Offset

1,2

Comments

a(n) = sum(k^2*A115994(n,k), k=1..floor(sqrt(n))).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Formula

G.f.: sum(k^2*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..infinity).

a(n) ~ sqrt(3) * (log(2))^2 * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - Vaclav Kotesovec, Jan 03 2019

Example

a(4) = 8 because the partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], have Durfee squares of sizes 1,1,2,1 and 1, respectively and 1^2+1^2+2^2+1^2+1^2=8.

Maple

g:=sum(k^2*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..10): gser:=series(g, z=0, 52): seq(coeff(gser, z^n), n=1..45);
# second Maple program:
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:
a:= n-> add(k^2*add(b(m, k)*b(n-k^2-m, k),
            m=0..n-k^2), k=1..floor(sqrt(n))):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 09 2012

Mathematica

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]]]]; a[n_] := Sum [k^2*Sum[b[m, k]*b[n - k^2 - m, k], {m, 0, n - k^2}], {k, 1, Sqrt[n]}]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

Crossrefs

Cf. A115994, A115995.

Sequence in context: A213046 A262021 A221181 * A291589 A105204 A352603

Adjacent sequences: A116500 A116501 A116502 * A116504 A116505 A116506

Keyword

easy,nonn

Author

Emeric Deutsch, Vladeta Jovovic, Feb 18 2006

Status

approved