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%I #46 Sep 08 2022 08:44:51
%S 120,2520,31920,317520,2739240,21538440,158838240,1118557440,
%T 7612364760,50483192760,328191186960,2100689987760,13282470124680,
%U 83169792213480,516729467446080,3190281535536480,19596640721427000,119876382958008600
%N Number of ways to partition n labeled elements into 6 pie slices.
%C 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
%H Vincenzo Librandi, <a href="/A032180/b032180.txt">Table of n, a(n) for n = 6..1000</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F "CIJ[ 6 ]" (necklace, indistinct, labeled, 6 parts) transform of 1, 1, 1, 1...
%F a(n) = 120*S(n, 6).
%F From _Emeric Deutsch_, May 02 2004: (Start)
%F a(n) = 5*2^(n-1) - 10*3^(n-1) + 10*4^(n-1) - 5^n + 6^(n-1) - 1.
%F a(n) = 120*A000770(n). (End)
%F 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
%F 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
%p 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
%t 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 *)
%t Table[120*StirlingS2[n,6], {n,6,30}] (* _G. C. Greubel_, Nov 19 2017 *)
%o (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
%o (PARI) for(n=6,30, print1(120*stirling(n,6,2), ", ")) \\ _G. C. Greubel_, Nov 19 2017
%Y Cf. A000770, A008277, A000225, A028243, A028244, A028245.
%K nonn,easy
%O 6,1
%A _Christian G. Bower_
%E More terms from _Vincenzo Librandi_ Oct 19 2013
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