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%I #24 Oct 24 2023 10:09:23
%S 1,2,45,43968,6358196250,236919104155855296,
%T 3682959509036574988532481464,
%U 35398008251644050232134479709365068115968,292415292106611727928759157427747328169866020125762652311
%N Number of n-element groupoids with an identity.
%C Also partial groupoids with n-1 elements or 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} ( (1 + sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j))
%F a(n) asymptotic to n^((n-1)^2+1)/n! = A090602(n)/A000142(n) = A090603(n)/A000142(n-1)
%o (Sage)
%o R.<a> = InfinitePolynomialRing(QQ)
%o @cached_function
%o def Z(n):
%o if n==0:
%o return R.one()
%o return sum(a[k]*Z(n-k) for k in (1..n))/n
%o def magmas_identity(n):
%o P = Z(n-1)
%o q = 0
%o c = P.coefficients()
%o count = 0
%o for m in P.monomials():
%o r = 1
%o T = m.variables()
%o S = list(T)
%o for u in T:
%o i = R.varname_key(str(u))[1]
%o j = m.degree(u)
%o D = 1
%o for d in divisors(i):
%o D += d*m.degrees()[-d-1]
%o r *= D^(i*j^2)
%o S.remove(u)
%o for v in S:
%o k = R.varname_key(str(v))[1]
%o l = m.degree(v)
%o D = 1
%o for d in divisors(lcm(i,k)):
%o try:
%o D += d*m.degrees()[-d-1]
%o except:
%o break
%o r *= D^(gcd(i,k)*j*l*2)
%o q += c[count]*r
%o count += 1
%o return q
%o # _Philip Turecek_, Oct 10 2023
%K nonn
%O 1,2
%A _Christian G. Bower_, Dec 05 2003
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