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G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-(tq)^j), j=1..k), k=1..inftyoo)}]_{t=1}.
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a(n) = Sum(_{k=0..n-1} k*A114087(n,k), k=0..n-1).
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 *) *)
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Vaclav Kotesovec, <a href="/A116365/b116365_1.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
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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