A052805 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.

1, 1, 1, 3, 7, 21, 64, 204, 666, 2236, 7625, 26419, 92644, 328370, 1174234, 4231898, 15354424, 56042372, 205626906, 758021598, 2806143522, 10427671924, 38882984840, 145443260702, 545598228056, 2052086677666, 7736986142773, 29236241424977

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn
• nice

Comments

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).

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 766

Crossrefs

Cf. A000108, A052854.

Programs

Maple

spec := [S,{C=Sequence(B),B=Prod(C,Z),S=PowerSet(B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

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 *)

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))

Formula

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

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).

G.f.: Product_{k>=1} (1+x^k)^(1/k*binomial(2*k-2, k-1)). - Vladeta Jovovic, Jan 17 2002