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A006448 Number of n-element algebras with 1 binary operator and 1 constant (pointed groupoids).
(Formerly M5029)
2

%I M5029 #29 Dec 19 2021 12:01:42

%S 1,16,9882,715860992,12417636281312500,85953540009068492207916672,

%T 356838302112667713247240882121025536249,

%U 1245456693529103515171728481423145699858332531028201472

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

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

%H M. A. Harrison, <a href="https://doi.org/10.1090/S0002-9939-1966-0200219-9">The number of isomorphism types of finite algebras</a>, Proc. Amer. Math. Soc., 17 (1966), 731-737.

%H <a href="/index/Gre#groupoids">Index entries for sequences related to groupoids</a>

%F 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]

%F 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)).

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

%Y Cf. A001329.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E Formula and more terms from _Christian G. Bower_, May 08 1998, Dec 03 2003

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)