A001425 Number of commutative groupoids with n elements. (Formerly M3714 N1518)

1, 1, 4, 129, 43968, 254429900, 30468670170912, 91267244789189735259, 8048575431238519331999571800, 24051927835861852500932966021650993560, 2755731922430783367615449408031031255131879354330, 13513302615133133128014689228630596983478739041461798638894834

Offset

0,3

References

Satoh, S.; Yama, K.; and Tokizawa, M., Semigroups of order 8, Semigroup Forum 49 (1994), 7-29. [Background]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..36

Eric Postpischil Posting to sci.math newsgroup, May 21 1990

N. J. A. Sloane, Overview of A001329, A001423-A001428, A258719, A258720.

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. (Annotated and scanned copy)

Index entries for sequences related to groupoids

Formula

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, ...] = Product_{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).

a(n) asymptotic to (n^binomial(n+1, 2))/n! = A023813(n)/A000142(n) ~ e^n*n^binomial(n, 2) / sqrt(2*pi*n).

a(n) = A001329(n) - A079183(n).

Prog

(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
C(d, r)={sum(i=1, #r, my(t=r[i]); if(d%t==0, t))}
E(v) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); C(v[i]*v[j]/g, v)^g)) * prod(i=1, #v, my(t=v[i]); C(t, v)^((t+1)\2)*if(t%2, 1, C(t/2, v)))}
a(n) = {my(s=0); forpart(p=n, s += permcount(p)*E(p)); s/n!} \\ Andrew Howroyd, Dec 10 2025

Crossrefs

Cf. A001329, A023813 (labeled version), A038016, A079183.

Sequence in context: A298274 A299136 A117897 * A050284 A096759 A299367

Adjacent sequences: A001422 A001423 A001424 * A001426 A001427 A001428

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower Feb 15 1998 and May 15 1998. Formula Dec 03 2003.

Status

approved