A032180 Number of ways to partition n labeled elements into 6 pie slices.

120, 2520, 31920, 317520, 2739240, 21538440, 158838240, 1118557440, 7612364760, 50483192760, 328191186960, 2100689987760, 13282470124680, 83169792213480, 516729467446080, 3190281535536480, 19596640721427000, 119876382958008600

Offset

6,1

Comments

For n>=6, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4,5,6} such that Im(f) contains 5 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Links

Vincenzo Librandi, Table of n, a(n) for n = 6..1000

C. G. Bower, Transforms (2)

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Index entries for sequences related to necklaces

Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Formula

"CIJ[ 6 ]" (necklace, indistinct, labeled, 6 parts) transform of 1, 1, 1, 1...

a(n) = 120*S(n, 6).

From Emeric Deutsch, May 02 2004: (Start)

a(n) = 5*2^(n-1) - 10*3^(n-1) + 10*4^(n-1) - 5^n + 6^(n-1) - 1.

a(n) = 120*A000770(n). (End)

G.f.: 120*x^6/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Sep 03 2012

E.g.f.: (Sum_{k=0..6} (-1)^(6-k)*binomial(6,k)*exp(k*x))/6 with a(n) = 0 for n = 0..5. - Wolfdieter Lang, May 03 2017

Maple

with (combstruct):ZL:=[S, {S=Sequence(U, card=r), U=Set(Z, card>=1)}, labeled]: seq(count(subs(r=6, ZL), size=m)/6, m=6..21); # Zerinvary Lajos, Mar 08 2008

Mathematica

CoefficientList[Series[120/((x - 1) (2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 19 2013 *)
Table[120*StirlingS2[n, 6], {n, 6, 30}] (* G. C. Greubel, Nov 19 2017 *)

Prog

(Magma) [5*2^(n-1)-10*3^(n-1)+10*4^(n-1)-5^n+6^(n-1)-1: n in [6..30]]; // Vincenzo Librandi, Oct 19 2013
(PARI) for(n=6, 30, print1(120*stirling(n, 6, 2), ", ")) \\ G. C. Greubel, Nov 19 2017

Crossrefs

Cf. A000770, A008277, A000225, A028243, A028244, A028245.

Sequence in context: A038745 A267839 A220050 * A000553 A126232 A105943

Adjacent sequences: A032177 A032178 A032179 * A032181 A032182 A032183

Keyword

nonn,easy

Author

Christian G. Bower

Extensions

More terms from Vincenzo Librandi Oct 19 2013

Status

approved