A133800 -id:A133800 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 17 2008

nonn

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

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

A diagonal of triangle A133800.

Cf. A212359.

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

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

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

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

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

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

A032262 Number of ways to partition n labeled elements into pie slices allowing the pie to be turned over.

Original entry on oeis.org

1, 1, 2, 5, 17, 83, 557, 4715, 47357, 545963, 7087517, 102248075, 1622633597, 28091569643, 526858352477, 10641342978635, 230283190994237, 5315654682014123, 130370767029201437, 3385534663256976395

Offset: 0

Christian G. Bower

nonn

			For n = 4 we have the following "pies":
. 1
./ \
2 . 3 . 12 .. 12 . 123 .1234
.\ / .. / \ .(..)..(..)
. 4 .. 3--4 . 34 .. 4
.(3)....(6)...(3)..(4)...(1) Total a(4) = 17

Andrew Howroyd, Table of n, a(n) for n = 0..200
C. G. Bower, Transforms (2)

Row sums of triangle A133800.

a[0] = a[1] = 1; a[n_] := 2^(n-2) + HurwitzLerchPhi[1/2, 1-n, 0]/2;
Array[a, 20, 0] (* Jean-François Alcover, Aug 26 2019 *)

seq(n)={my(p=exp(x + O(x*x^n))-1); Vec(1 + serlaplace(p + p^2/2 - log(1-p))/2)} \\ Andrew Howroyd, Sep 12 2018

a(n) = 2^(n-2) + A000670(n-1) for n >= 2. - N. J. A. Sloane, Jan 17 2008
a(n) = 2^(n-1) + Sum_{k >= 3} Stirling_2(n,k)*(k-1)!/2 for n >= 1. - N. J. A. Sloane, Jan 17 2008
"DIJ" (bracelet, indistinct, labeled) transform of 1, 1, 1, 1, ... (see Bower link).
E.g.f.: 1 + (g(x) + g(x)^2/2 - log(1-g(x)))/2 where g(x) = exp(x) - 1. - Andrew Howroyd, Sep 12 2018

Edited by N. J. A. Sloane, Jan 17 2008

a(0)=1 prepended by Andrew Howroyd, Sep 12 2018

cp's OEIS Frontend

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

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