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%I #15 Dec 19 2021 13:43:21
%S 1,2,15,720,409600,3920030472,775775333825891,3837862827737186253664,
%T 558740081065710564284870598075,
%U 2755731923933734753149997221152548428020,520996314135332606285488148844494695722050333912483
%N Number of n-element commutative groupoids with an identity ("pointed" groupoids).
%C Also number of commutative partial groupoids with n-1 elements or commutative groupoids with an absorbant (zero) element with n elements.
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Groupoid.html">Groupoid.</a>
%H <a href="/index/Gre#groupoids">Index entries for sequences related to groupoids</a>
%F a(n+1) = 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} (1 + sum {d|i} (d*s_d))^((i*s_i^2+s_i)/2) or {i=j, even} (1 + sum {d|i} (d*s_d))^(i*s_i^2/2) * (1 + sum {d|i/2} (d*s_d))^s_i or {i != j} (1 + sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)
%F a(n) asymptotic to (n^binomial(n, 2)+1)/n! = A090599(n)/A000142(n) = A076113(n)/A000142(n-1)
%Y Cf. A001329, A030257.
%K nonn
%O 1,2
%A _Christian G. Bower_, May 15 1998; revised Dec 05 2003
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