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A133799 a(2) = 1, a(3)=3; for n >= 4, a(n) = (n-2)!*Stirling_2(n,n-1)/2 = n!/4.

%I A133799 #29 Sep 28 2024 05:35:49
%S A133799 1,3,6,30,180,1260,10080,90720,907200,9979200,119750400,1556755200,
%T A133799 21794572800,326918592000,5230697472000,88921857024000,
%U A133799 1600593426432000,30411275102208000,608225502044160000,12772735542927360000,281000181944401920000
%N A133799 a(2) = 1, a(3)=3; for n >= 4, a(n) = (n-2)!*Stirling_2(n,n-1)/2 = n!/4.
%C A133799 a(n-1), n>=5, gives the number of necklaces with n beads (C_n symmetry) with color signature determined from the partition 2^2,1^(n-4) of n. Only n-2 distinct colors, say c[1], c[2], ..., c[n-2] are used, and the representative necklaces have the color c[1] and c[2] each twice. E.g., n=5, partition 2,2,1, color signature (take the parts as exponents) c[1]c[1]c[2]c[2]c[3], with the a(4)=6 necklaces (write j for color c[j]) 11223, 11232, 11322, 12213, 12123 and 12132,  all taken cyclically. See A212359 for the numbers for general partitions or color signatures. - _Wolfdieter Lang_, Jun 27 2012
%H A133799 Vincenzo Librandi, <a href="/A133799/b133799.txt">Table of n, a(n) for n = 2..200</a>
%F A133799 a(n) = numerator(n!/(2*(n! - 2))) for n > 2. - _Stefano Spezia_, Dec 06 2023
%t A133799 Join[{1,3},Range[4,30]!/4] (* _Harvey P. Dale_, Aug 13 2013 *)
%o A133799 (PARI) concat([1,3],vector(66,n,(n+3)!/4)) \\ _Joerg Arndt_, Aug 14 2013
%o A133799 (Magma) [n le 3 select 2*n-3 else Factorial(n)/4: n in [2..30]]; // _G. C. Greubel_, Sep 28 2024
%o A133799 (SageMath)
%o A133799 def A133799(n): return (factorial(n) +2*int(n==2) +6*int(n==3))//4
%o A133799 [A133799(n) for n in range(2,31)] # _G. C. Greubel_, Sep 28 2024
%Y A133799 A diagonal of triangle A133800.
%Y A133799 Cf. A212359.
%K A133799 nonn
%O A133799 2,2
%A A133799 _N. J. A. Sloane_, Jan 17 2008
%E A133799 Corrected parameters in definition. - _Geoffrey Critzer_, Apr 26 2009