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

40320, 1451520, 30240000, 479001600, 6411968640, 76592355840, 843184742400, 8734434508800, 86355926616960, 823172919528960, 7621934141203200, 68937160460313600, 611692004959217280, 5342844138794426880, 46061530905262118400, 392795626402384128000

Offset: 8

Author

Mohamed Bouhamida, Dec 16 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 8..1000
• Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Crossrefs

Column k=8 of A019538 and A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A135456.

Programs

Magma

[&+[(-1)^(8-k)*Binomial(8, k)*k^n: k in [1..n]]: n in [8..25]]; // Vincenzo Librandi, Oct 21 2017

Mathematica

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 *)
Table[Sum[(-1)^(8 - k)*Binomial[8, k]*k^n, {k, 1, 8}], {n, 8, 20}] (* G. C. Greubel, Oct 21 2017 *)

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

Formula

a(n) = Sum_{k=1..8} ((-1)^(8-k)*binomial(8,k)*k^n).

a(n) = A049434(n) * 8!. - Max Alekseyev, Nov 13 2009

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

E.g.f.: (exp(x) - 1)^8. - Ilya Gutkovskiy, Jun 19 2018

Extensions

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

More terms from Max Alekseyev, Nov 13 2009