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