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

Author

Emeric Deutsch, Vladeta Jovovic, Feb 18 2006

Keywords

• easy
• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A115994, A115995.

Programs

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 *)

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