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

%I A116503 #21 Jan 03 2019 08:04:59
%S A116503 1,2,3,8,13,26,39,64,98,148,216,322,455,648,904,1258,1711,2336,3128,
%T A116503 4198,5548,7330,9569,12496,16146,20836,26674,34098,43273,54846,69072,
%U A116503 86848,108627,135612,168527,209066,258271,318482,391321,479946,586709
%N A116503 Sum of the areas of the Durfee squares of all partitions of n.
%C A116503 a(n) = sum(k^2*A115994(n,k), k=1..floor(sqrt(n))).
%H A116503 Vaclav Kotesovec, <a href="/A116503/b116503.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A116503 G.f.: sum(k^2*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..infinity).
%F A116503 a(n) ~ sqrt(3) * (log(2))^2 * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - _Vaclav Kotesovec_, Jan 03 2019
%e A116503 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.
%p A116503 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);
%p A116503 # second Maple program:
%p A116503 b:= proc(n, i) option remember;
%p A116503       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A116503     end:
%p A116503 a:= n-> add(k^2*add(b(m, k)*b(n-k^2-m, k),
%p A116503             m=0..n-k^2), k=1..floor(sqrt(n))):
%p A116503 seq(a(n), n=1..40);  # _Alois P. Heinz_, Apr 09 2012
%t A116503 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_ *)
%Y A116503 Cf. A115994, A115995.
%K A116503 easy,nonn
%O A116503 1,2
%A A116503 _Emeric Deutsch_, _Vladeta Jovovic_, Feb 18 2006