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%I #29 Jan 03 2019 06:13:05
%S 1,1,1,3,3,7,7,13,16,24,30,46,55,79,100,136,169,229,282,374,462,598,
%T 737,947,1158,1466,1794,2246,2733,3399,4116,5076,6133,7503,9033,10993,
%U 13177,15943,19061,22939,27327,32749,38883,46395,54938,65278,77070,91270
%N Sum of the sizes of the Durfee squares of all partitions of n that do not contain 1 as a part, but with a(1) = 1.
%C Also sum of the sizes of the Durfee squares of all partitions of the head of the last section of n (see A135010).
%H Alois P. Heinz, <a href="/A208474/b208474.txt">Table of n, a(n) for n = 1..1000</a>
%H G. E. Andrews, <a href="http://www.math.psu.edu/andrews/pdf/80.pdf">Partitions and Durfee Dissection</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DurfeeSquare.html">Durfee Square</a>
%F a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - _Vaclav Kotesovec_, Jan 03 2019
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p end:
%p g:= proc(n) option remember;
%p add(add(b(k, d)*b(n-d^2-k, d),
%p k=0..n-d^2)*d, d=1..floor(sqrt(n)))
%p end:
%p a:= n-> g(n)-g(n-1):
%p seq(a(n), n=1..70); # _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]]]]; g[n_] := Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Sqrt[n]}]; Table[g[n], {n, 0, 70}] // Differences (* _Jean-François Alcover_, Feb 21 2017, after _Alois P. Heinz_ *)
%Y First differences of A115995.
%Y Cf. A135010, A138135, A138137, A138879.
%K nonn
%O 1,4
%A _Omar E. Pol_, Mar 03 2012
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