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FORMULA
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product(1+a(n)*x^n,n=1..infinity) = sum(C(k)*x^k,k=1..infinity) = (1-sqrt(1-4*x))/(2*x), with C(n)= A000108(n) (Catalan numbers).
Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)):
a(n)= C(n) - sum(sum(product(a(k[j]),j=1..m), fp from FP(n,m)), m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=C(1)=1, a(2)=C(2)=2. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: With P(n,m) the set of all partitions of n with m parts, and the multinomial numbers M0 (given for every partition under A048996):
a(n) = sum((d/n)*(-a(d)^(n/d)),d|n with 1<d<n) + sum(((-1)^(m-1))*(1/m)*sum(M0(p)*C(1)^e(1)*...*C(n)^e(n), p=(1^e(1),...,n^e(n)) from P(n,m)), m=1..n-1), n>=2; a(1)=C(1)=1. The exponents e(j)>=0 satisfy sum(j*e(j),j=1..n)=n and sum(e(j),j=1..m). If e_j=0 then part j does not appear. The M0 numbers are m!/product(e(j)!,j=1..n).
Recurrence II (rewritten, due email from V. Jovovic, Mar 10 2009):
a(n)= (sum((d/n)*(-a(d))^(n/d),d|n with 1<=d<n) + (2*n-1)!/n!^2, n>=2; a(1)=1. Note that n*(2*n-1)!/n!^2 = A001700(n-1)= A088218(n), n>=1, with o.g.f. diff(ln(c(x)),x), where c(x) is the o.g.f. for Catalan numbers A000108. Here no partitions are needed.
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