A001330 Number of n-element algebras with 2 binary operations.

1, 1, 136, 64573605, 768614338015543296, 740148683083442627372862307855625, 147760220727384062234340471228346859265417269763446784, 13097167596472133103922286145973062271265962292695709182416029922453889335720758

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Isomorphisms classes of a set A with two functions f1,f2: A X A -> A.

References

• 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).

Links

• M. A. Harrison, The number of isomorphism types of finite algebras, Proc. Amer. Math. Soc., 17 (1966), 731-737.

Crossrefs

Cf. A001329, A001331.

Formula

a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = Product_{i, j>=1} ( (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j*2)).

a(n) is asymptotic to n^(2*n^2)/n! = A008972(n)/A000142(n).

Extensions

Edited and extended with formula by Christian G. Bower, Jan 06 2004