OFFSET
0,3
COMMENTS
For n>=1, a(n) is equal to the number of surjections from {1,2,...,n+4} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007
REFERENCES
Identity (1.20) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 3.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..350
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
FORMULA
From G. C. Greubel, Jun 20 2022: (Start)
a(n) = (-1)^n * Sum_{j=0..n} (-1)^j * binomial(n, j)*j^(n+4).
a(n) = n!*StirlingS2(n+4, n).
a(n) = A131689(n+4, n).
a(n) = A019538(n+4, n).
E.g.f.: x*(1 + 22*x + 58*x^2 + 24*x^3)/(1-x)^9. (End)
MATHEMATICA
Table[(n+4)!n(15n^3+30n^2+5n-2)/5760, {n, 0, 20}] (* Harvey P. Dale, Nov 16 2020 *)
Table[n!*StirlingS2[n+4, n], {n, 0, 30}] (* G. C. Greubel, Jun 20 2022 *)
PROG
(Magma) [Factorial(n)*StirlingSecond(n+4, n): n in [0..30]]; // G. C. Greubel, Jun 20 2022
(SageMath) [factorial(n)*stirling_number2(n+4, n) for n in (0..30)] # G. C. Greubel, Jun 20 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved