A028245 a(n) = 5^(n-1) - 4*4^(n-1) + 6*3^(n-1) - 4*2^(n-1) + 1 (essentially Stirling numbers of second kind).

0, 0, 0, 0, 24, 360, 3360, 25200, 166824, 1020600, 5921520, 33105600, 180204024, 961800840, 5058406080, 26308573200, 135666039624, 694994293080, 3542142833040, 17980946172000, 90990301641624

Offset: 1

Author

N. J. A. Sloane, Doug McKenzie mckfam4(AT)aol.com

Keywords

• nonn
• easy

Comments

For n>=2, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4,5} such that Im(f) contains 4 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1431
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Crossrefs

Cf. A000481, A008277, A163626, A000225, A028243, A028244.

Programs

Magma

[5^(n-1) - 4*4^(n-1) + 6*3^(n-1) - 4*2^(n-1) + 1: n in [1..30]]; // G. C. Greubel, Nov 19 2017

Mathematica

24StirlingS2[Range[30],5] (* Harvey P. Dale, Jun 18 2013 *)
Table[5^(n - 1) - 4*4^(n - 1) + 6*3^(n - 1) - 4*2^(n - 1) + 1, {n, 21}] (* or *)
Rest@ CoefficientList[Series[-24 x^5/((x - 1) (4 x - 1) (3 x - 1) (2 x - 1) (5 x - 1)), {x, 0, 21}], x] (* Michael De Vlieger, Sep 24 2016 *)

PARI

for(n=1,30, print1(24*stirling(n,5,2), ", ")) \\ G. C. Greubel, Nov 19 2017

Formula

a(n) = 24*S(n, 5) = 24*A000481(n). - Emeric Deutsch, May 02 2004

G.f.: -24*x^5/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009

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