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Order of the subgroup of the symmetric group S_n generated by the cycles (1,2,3) and (1,2,3,...,n).
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%I #11 Jul 06 2022 06:57:57

%S 3,24,60,720,2520,40320,181440,3628800,19958400,479001600,3113510400,

%T 87178291200,653837184000,20922789888000,177843714048000,

%U 6402373705728000,60822550204416000,2432902008176640000

%N Order of the subgroup of the symmetric group S_n generated by the cycles (1,2,3) and (1,2,3,...,n).

%F For odd n: a(n) = n!/2; for even n: a(n) = n!.

%F a(n) = (1/4)*(3+(-1)^n)*n! - _Benoit Cloitre_, May 18 2002

%F From _Amiram Eldar_, Jul 06 2022: (Start)

%F Sum_{n>=3} 1/a(n) = 2*sinh(1) + cosh(1) - 7/2.

%F Sum_{n>=3} (-1)^(n+1)/a(n) = 2*sinh(1) - cosh(1) - 1/2. (End)

%t f[n_] := If[ EvenQ[n], n!, n!/2]; Table[ f[n], {n, 3, 24}].

%o (PARI) for(n=3,20,print1((3+(-1)^n)/4*n!,","))

%Y Cf. A000142, A001710.

%K nonn,easy

%O 3,1

%A Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

%E More terms from _Benoit Cloitre_, May 18 2002