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A123114 a(n) = Sum_{r>0,s>0} binomial(r*s-1,n-1)/2^(r+s).

%I A123114 #13 Aug 09 2022 00:15:28
%S A123114 1,3,13,83,701,7363,92541,1354627,22636861,425241347,8871085565,
%T A123114 203487078403,5090418231549,137920771272963,4023549748488445,
%U A123114 125743894742698243,4191213031967650813,148414827031140706307
%N A123114 a(n) = Sum_{r>0,s>0} binomial(r*s-1,n-1)/2^(r+s).
%H A123114 Vaclav Kotesovec, <a href="/A123114/b123114.txt">Table of n, a(n) for n = 1..400</a>
%H A123114 M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">On the arithmetic product of combinatorial species</a>
%F A123114 a(n) = (1/(n-1)!)*Sum_{k=1..n} Stirling1(n,k)*b(k)^2, where b(n) = Sum_{k=1..n} (k-1)!*Stirling2(n,k).
%F A123114 a(n) ~ c * (n-1)! / (log(2))^(2*n), where c = 2^(-log(2)/2) = 0.7864497045594053649114085152934509198700275589579678941719548714254307448... - _Vaclav Kotesovec_, Jun 07 2019
%t A123114 Table[Sum[StirlingS1[n, k]*(Sum[(j - 1)!*StirlingS2[k, j], {j, 1, k}])^2, {k, 1, n}]/(n-1)!, {n, 1, 20}] (* _Vaclav Kotesovec_, Jun 07 2019 *)
%t A123114 Table[-(-1)^n + Sum[StirlingS1[n, k]*PolyLog[1-k, 2]^2, {k, 2, n}]/(n-1)!, {n, 1, 20}] (* _Vaclav Kotesovec_, Jun 07 2019 *)
%Y A123114 Cf. A101370, A000629.
%K A123114 easy,nonn
%O A123114 1,2
%A A123114 _Vladeta Jovovic_, Sep 28 2006