A006448 Number of n-element algebras with 1 binary operator and 1 constant (pointed groupoids).

1, 16, 9882, 715860992, 12417636281312500, 85953540009068492207916672, 356838302112667713247240882121025536249, 1245456693529103515171728481423145699858332531028201472

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

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

Links

• M. A. Harrison, The number of isomorphism types of finite algebras, Proc. Amer. Math. Soc., 17 (1966), 731-737.
• Index entries for sequences related to groupoids

Crossrefs

Cf. A001329.

Formula

For a list n(1), n(2), n(3), ..., let fixF[n] = n(1)*Product_{i, j >= 1} (Sum_{d|[ i, j ]} (d*n(d))^((i, j)*n(i)*n(j))), where [i,j] = lcm(i,j). [Note that the notation fixF[n] appears in several other formulas contributed by Christian G. Bower. In this case it seems that the prefix "fix" was accidentally removed over the course of the years. - N. J. A. Sloane, Dec 19 2021]

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, ...] = s_1 * Product_{i, j>=1} ( (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)).

a(n) is asymptotic to n^(n^2+1)/n!.

Extensions

Formula and more terms from Christian G. Bower, May 08 1998, Dec 03 2003