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A116365 Sum of the sizes of the tails below the Durfee squares of all partitions of n.

%I A116365 #27 Sep 11 2024 10:08:31
%S A116365 0,1,3,6,11,20,33,56,86,136,200,301,429,621,868,1219,1669,2297,3091,
%T A116365 4171,5542,7357,9648,12652,16402,21250,27298,35003,44556,56637,71515,
%U A116365 90160,113046,141464,176189,219053,271149,335044,412447,506787,620597
%N A116365 Sum of the sizes of the tails below the Durfee squares of all partitions of n.
%D A116365 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
%D A116365 G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
%H A116365 Vaclav Kotesovec, <a href="/A116365/b116365.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A116365 a(n) = Sum_{k=0..n-1} k*A114087(n,k).
%F A116365 G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-(tq)^j), j=1..k), k=1..oo)}]_{t=1}.
%F A116365 a(n) = (n*A000041(n)-A116503(n))/2. - _Vladeta Jovovic_, Feb 18 2006
%F A116365 a(n) ~ (1/(8*sqrt(3)) - sqrt(3) * (log(2))^2 / (4*Pi^2)) * exp(Pi*sqrt(2*n/3)). - _Vaclav Kotesovec_, Jan 03 2019
%e A116365 a(4) = 6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having total size 0+1+0+2+3=6.
%p A116365 g:=sum(z^(k^2)/product((1-z^j)*(1-(t*z)^j),j=1..k),k=1..10): dgdt1:=simplify(subs(t=1,diff(g,t))): dgdt1ser:=series(dgdt1,z=0,55): seq(coeff(dgdt1ser,z,n),n=1..48);
%p A116365 # second Maple program:
%p A116365 b:= proc(n, i) option remember;
%p A116365       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A116365     end:
%p A116365 a:= n-> add(k*add(b(k, d) *b(n-d^2-k, d),
%p A116365             d=0..floor(sqrt(n))), k=0..n-1):
%p A116365 seq(a(n), n=1..40);  # _Alois P. Heinz_, Apr 2012
%t A116365 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*Sum[b[k, d]*b[n-d^2-k, d], {d, 0, Floor[Sqrt[n]]}], {k, 0, n-1}]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Mar 31 2015, after _Alois P. Heinz_ *)
%Y A116365 Cf. A115994, A115995, A114087, A114088, A114089.
%K A116365 nonn
%O A116365 1,3
%A A116365 _Emeric Deutsch_, Feb 12 2006