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A362382 Number of nonisomorphic right involutory magmas with n elements.

%I A362382 #13 Apr 18 2023 14:45:33
%S A362382 1,1,3,16,475,100666,267954164,7178089200724,2878905036230723360,
%T A362382 16030557330452794172050567,1643024454743084814743097053747492,
%U A362382 3003719433250221394022136941323628209106412,119909786948816191249293422143299520925389900896422044
%N A362382 Number of nonisomorphic right involutory magmas with n elements.
%C A362382 A magma with element set X is right involutory if (xy)y = x for x,y in X.
%H A362382 Andrew Howroyd, <a href="/A362382/b362382.txt">Table of n, a(n) for n = 0..40</a>
%o A362382 (PARI)
%o A362382 B(c,k)=sum(j=0, c\2, if(k%2, 1, 2^(c-2*j))*k^j*binomial(c, 2*j)*(2*j)!/(2^j*j!))
%o A362382 K(v)=my(S=Set(v)); prod(i=1, #S, my(k=S[i], c=#select(t->t==k, v)); B(c,k))
%o A362382 R(v,m)=concat(vector(#v,i,my(t=v[i], g=gcd(t,m)); vector(g, i, t/g)))
%o A362382 a(n)={my(s=0); forpart(p=n, my(v=Vec(p), S=Set(v)); s+=prod(i=1, #S, my(m=S[i], c=#select(t->t==m, v)); (K(R(v,m))/m)^c/c!)); s}
%Y A362382 Cf. A001329 (magmas), A076017, A076019, A361720, A362383 (labeled).
%K A362382 nonn
%O A362382 0,3
%A A362382 _Andrew Howroyd_, Apr 17 2023