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A194924 The number of set partitions of {1,2,...,n} into exactly two subsets A,B such that the greatest common divisor of |A| and |B| = 1.

%I A194924 #25 Jun 09 2021 05:24:33
%S A194924 1,3,4,15,6,63,64,171,130,1023,804,4095,2380,7920,16384,65535,40410,
%T A194924 262143,246640,582771,695860,4194303,2884776,13455325,11576916,
%U A194924 44739243,65924824,268435455,176422980,1073741823,1073741824,2669774811,3128164186,11421338075
%N A194924 The number of set partitions of {1,2,...,n} into exactly two subsets A,B such that the greatest common divisor of |A| and |B| = 1.
%C A194924 a(p)=2^(p-1)-1 = S2(p,2) where p is a prime and S2(n,k) is the Stirling number of the second kind.
%C A194924 a(n) is the coefficient of x^n/n! in the Taylor series expansion of B(A(x)) where A(x)= Sum_{over positive integers relatively prime to n}x^n/n! and B(x)=x^2/2!.
%H A194924 Alois P. Heinz, <a href="/A194924/b194924.txt">Table of n, a(n) for n = 2..1000</a>
%p A194924 a:= n-> `if`(n=2, 1, add(`if`(igcd(k, n-k)=1,
%p A194924                      binomial(n, k), 0), k=1..iquo(n, 2))):
%p A194924 seq(a(n), n=2..50); # _Alois P. Heinz_, Nov 02 2011
%t A194924 f[list_]:=x^First[list]/First[list]!+x^Last[list]/Last[list]!;
%t A194924 Prepend[Table[a=Total[Map[f,Select[IntegerPartitions[n,2],Apply[GCD,#]==1&]]];Last[Range[0,n]! CoefficientList[Series[a^2/2!,{x,0,n}],x]],{n,3,30}],1]
%t A194924 (* Second program: *)
%t A194924 a[n_] := If[n == 2, 1, Sum[If[GCD[k, n-k] == 1, Binomial[n, k], 0], {k, 1, Quotient[n, 2]}]];
%t A194924 a /@ Range[2, 50] (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)
%Y A194924 Column k=2 of A280880.
%K A194924 nonn
%O A194924 2,2
%A A194924 _Geoffrey Critzer_, Oct 12 2011