%I A001425 M3714 N1518
%S A001425 1,1,4,129,43968,254429900,30468670170912,91267244789189735259,
%T A001425 8048575431238519331999571800,24051927835861852500932966021650993560,
%U A001425 2755731922430783367615449408031031255131879354330
%N A001425 Number of commutative groupoids with n elements.
%D A001425 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001425 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001425 Satoh, S.; Yama, K.; and Tokizawa, M., Semigroups of order 8, Semigroup 
               Forum 49 (1994), 7-29. [Background]
%D A001425 T. Tamura, Some contributions of computation to semigroups and groupoids, 
               pp. 229-261 of J. Leech, editor, Computational Problems in Abstract 
               Algebra. Pergamon, Oxford, 1970.
%H A001425 Eric Postpischil <a href="http://groups.google.com/groups?&hl=en&lr=&ie=UTF-8&selm=11802%40shlump.nac.dec.com\
               &rnum=2">Posting to sci.math newsgroup, May 21 1990</a>
%H A001425 <a href="Sindx_Gre.html#groupoids">Index entries for sequences related 
               to groupoids</a>
%F A001425 a(n) = sum {1*s_1+2*s_2+...=n} (fix A[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) 
               where fix A[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where 
               f(i, j, s_i, s_j) = {i=j, odd} (sum {d|i} (d*s_d))^((i*s_i^2+s_i)/
               2) or {i=j, even} (sum {d|i} (d*s_d))^(i*s_i^2/2) * (sum {d|i/2} 
               (d*s_d))^s_i or {i != j} (sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)
%F A001425 a(n) asymptotic to (n^binomial(n+1, 2))/n! = A023813(n)/A000142(n) ~ 
               e^n*n^binomial(n, 2) / sqrt(2*pi*n).
%Y A001425 a(n)+A079183(n)=A001329(n)
%Y A001425 Cf. A001329, A023813, A038016.
%Y A001425 Sequence in context: A057134 A041495 A117897 this_sequence A050284 A096759 
               A006103
%Y A001425 Adjacent sequences: A001422 A001423 A001424 this_sequence A001426 A001427 
               A001428
%K A001425 nonn
%O A001425 0,3
%A A001425 N. J. A. Sloane (njas(AT)research.att.com).
%E A001425 More terms from Christian G. Bower (bowerc(AT)usa.net) Feb 15 1998 and 
               May 15 1998. Formula Dec 03 2003.