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
KEYWORD
easy,nonn
AUTHOR
Emeric Deutsch, Vladeta Jovovic, Feb 18 2006
STATUS
approved