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A001425 Number of commutative groupoids with n elements.

%I A001425 M3714 N1518 #34 Jul 08 2025 16:26:22
%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 Satoh, S.; Yama, K.; and Tokizawa, M., Semigroups of order 8, Semigroup Forum 49 (1994), 7-29. [Background]
%D A001425 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001425 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%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?&amp;hl=en&amp;lr=&amp;ie=UTF-8&amp;selm=11802%40shlump.nac.dec.com&amp;rnum=2">Posting to sci.math newsgroup, May 21 1990</a>
%H A001425 N. J. A. Sloane, <a href="/A001329/a001329.jpg">Overview of A001329, A001423-A001428, A258719, A258720.</a>
%H A001425 T. Tamura, <a href="/A001329/a001329.pdf">Some contributions of computation to semigroups and groupoids</a>, pp. 229-261 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970. (Annotated and scanned copy)
%H A001425 <a href="/index/Gre#groupoids">Index entries for sequences related to groupoids</a>
%F A001425 a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[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))^(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.
%K A001425 nonn
%O A001425 0,3
%A A001425 _N. J. A. Sloane_
%E A001425 More terms from _Christian G. Bower_ Feb 15 1998 and May 15 1998. Formula Dec 03 2003.