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A133068 Number of surjections from an n-element set to an eight-element set.

%I A133068 #35 Mar 16 2024 17:25:23
%S A133068 40320,1451520,30240000,479001600,6411968640,76592355840,843184742400,
%T A133068 8734434508800,86355926616960,823172919528960,7621934141203200,
%U A133068 68937160460313600,611692004959217280,5342844138794426880,46061530905262118400,392795626402384128000
%N A133068 Number of surjections from an n-element set to an eight-element set.
%H A133068 G. C. Greubel, <a href="/A133068/b133068.txt">Table of n, a(n) for n = 8..1000</a>
%H A133068 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).
%F A133068 a(n) = Sum_{k=1..8} ((-1)^(8-k)*binomial(8,k)*k^n).
%F A133068 a(n) = A049434(n) * 8!. - _Max Alekseyev_, Nov 13 2009
%F A133068 G.f.: 40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - _Colin Barker_, Oct 25 2012
%F A133068 E.g.f.: (exp(x) - 1)^8. - _Ilya Gutkovskiy_, Jun 19 2018
%t A133068 CoefficientList[Series[40320*x^8/((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(7*x - 1)*(8*x - 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 20 2017 *)
%t A133068 Table[Sum[(-1)^(8 - k)*Binomial[8, k]*k^n, {k, 1, 8}], {n, 8, 20}] (* _G. C. Greubel_, Oct 21 2017 *)
%o A133068 (PARI) x='x+O('x^50); Vec(40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1))) \\ _G. C. Greubel_, Oct 20 2017
%o A133068 (Magma) [&+[(-1)^(8-k)*Binomial(8, k)*k^n: k in [1..n]]: n in [8..25]]; // _Vincenzo Librandi_, Oct 21 2017
%Y A133068 Column k=8 of A019538 and A131689.
%Y A133068 Cf. A000918, A000919, A000920, A001117, A001118, A135456.
%K A133068 nonn,easy
%O A133068 8,1
%A A133068 _Mohamed Bouhamida_, Dec 16 2007
%E A133068 Edited by _N. J. A. Sloane_, Jul 12 2008 at the suggestion of _R. J. Mathar_
%E A133068 More terms from _Max Alekseyev_, Nov 13 2009