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

1, 3, 6, 30, 180, 1260, 10080, 90720, 907200, 9979200, 119750400, 1556755200, 21794572800, 326918592000, 5230697472000, 88921857024000, 1600593426432000, 30411275102208000, 608225502044160000, 12772735542927360000, 281000181944401920000

Offset: 2

Author

N. J. A. Sloane, Jan 17 2008

Keywords

• nonn

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..200

Crossrefs

A diagonal of triangle A133800.

Cf. A212359.

Programs

Magma

[n le 3 select 2*n-3 else Factorial(n)/4: n in [2..30]]; // G. C. Greubel, Sep 28 2024

Mathematica

Join[{1,3},Range[4,30]!/4] (* Harvey P. Dale, Aug 13 2013 *)

PARI

concat([1,3],vector(66,n,(n+3)!/4)) \\ Joerg Arndt, Aug 14 2013

SageMath

def A133799(n): return (factorial(n) +2*int(n==2) +6*int(n==3))//4
[A133799(n) for n in range(2,31)] # G. C. Greubel, Sep 28 2024

Formula

a(n) = numerator(n!/(2*(n! - 2))) for n > 2. - Stefano Spezia, Dec 06 2023

Extensions

Corrected parameters in definition. - Geoffrey Critzer, Apr 26 2009