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A362383 Number of labeled right involutory magmas with n elements.

%I A362383 #16 Mar 14 2025 09:41:10
%S A362383 1,1,4,64,10000,11881376,192699928576,36175612601171968,
%T A362383 116077185312503648813056,5817168207073186596352000000000,
%U A362383 5962207128673051739782035558293177368576,119898867867315010793162270409575082620582830800896,57436979804085599487337333419576950752550097125586310052970496
%N A362383 Number of labeled right involutory magmas with n elements.
%C A362383 A magma with element set X is right involutory if (xy)y = x for x,y in X.
%H A362383 Andrew Howroyd, <a href="/A362383/b362383.txt">Table of n, a(n) for n = 0..30</a>
%F A362383 a(n) = A000085(n)^n.
%F A362383 a(n) ~ exp(-7/48 + 7*sqrt(n)/24 - n/4 + n^(3/2) - n^2/2) * n^(n^2/2) / 2^(n/2). - _Vaclav Kotesovec_, Mar 14 2025
%p A362383 b:= proc(n) b(n):= `if`(n<2, 1, b(n-1)+(n-1)*b(n-2)) end:
%p A362383 a:= n-> b(n)^n:
%p A362383 seq(a(n), n=0..15);  # _Alois P. Heinz_, Apr 30 2023
%t A362383 A85[n_] := Sum[StirlingS1[n, k]*2^k*BellB[k, 1/2], {k, 0, n}];
%t A362383 a[n_] := A85[n]^n;
%t A362383 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 14 2025, after _Emanuele Munarini_ in A85 *)
%o A362383 (PARI) \\ here b(n) is A000085(n).
%o A362383 b(n)=sum(j=0, n\2, binomial(n, 2*j)*(2*j)!/(2^j*j!))
%o A362383 a(n)=b(n)^n
%Y A362383 Cf. A000085, A002489 (magmas), A076016, A362382 (isomorphism classes).
%K A362383 nonn
%O A362383 0,3
%A A362383 _Andrew Howroyd_, Apr 17 2023