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%I #16 Apr 18 2017 07:04:08
%S 1,1,1,3,7,21,64,204,666,2236,7625,26419,92644,328370,1174234,4231898,
%T 15354424,56042372,205626906,758021598,2806143522,10427671924,
%U 38882984840,145443260702,545598228056,2052086677666,7736986142773,29236241424977
%N If B is a collection in which there are C(n-1) [Catalan numbers, A000108] things with n points, a(n) is the number of subsets without repetition of B with a total of n points.
%C Euler transform of sequence [1,0,2,4,14,40,132,424,1430,...] (C(n-1) if n odd, C(n-1)-C(n/2-1) if n even).
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=766">Encyclopedia of Combinatorial Structures 766</a>
%F a(n)=(1/n)*Sum_{k=1..n} a(n-k)*b(k), n>0, a(0)=1, b(k)=Sum_{d|k} (-1)^(k/d+1)*binomial(2*d-2, d-1). - _Vladeta Jovovic_, Jan 17 2002
%F G.f. A(x)=exp(Sum_{k>0} -(-1)^k* C(x^k)/k) where C(x)=(1-sqrt(1-4x))/2= g.f. A000108 (offset 1).
%F G.f.: Product_{k>=1} (1+x^k)^(1/k*binomial(2*k-2, k-1)). - _Vladeta Jovovic_, Jan 17 2002
%p spec := [S,{C=Sequence(B),B=Prod(C,Z),S=PowerSet(B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t ClearAll[a]; b[k_] := Sum[ (-1)^(k/d + 1)*Binomial[2*d - 2, d - 1], {d, Divisors[k]}]; a[0] = 1; a[n_] := a[n] = (1/n)*Sum[a[n - k]*b[k], {k, 1, n}]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Oct 08 2012, after _Vladeta Jovovic_ *)
%o (PARI) a(n)=local(A); if(n<1,!n,A=sum(k=1,n,(2*k-2)!/k!/(k-1)!*x^k,x*O(x^n)); polcoeff(exp(sum(k=1,n,-(-1)^k*subst(A,x,x^k)/k)),n))
%Y Cf. A000108, A052854.
%K easy,nonn,nice
%O 0,4
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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