OFFSET
0,3
COMMENTS
For n>=1, a(n) is equal to the number of surjections from {1,2,...,n+3} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007
REFERENCES
Identity (1.19) in H. W. Gould, Combinatorial Identities, Morgantown, 1972; page 3.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010 (section 10, p.45)
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
FORMULA
(n-1)^2*a(n) - n*(n+3)*(n+1)*a(n-1) = 0. - R. J. Mathar, Jul 26 2015
E.g.f.: x*(1 + 8*x + 6*x^2)/(1 - x)^7. - Ilya Gutkovskiy, Feb 20 2017
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*k^(n+3). - Peter Bala, Mar 28 2017
From G. C. Greubel, Jun 20 2022: (Start)
a(n) = n!*StirlingS2(n+3, n).
a(n) = A131689(n+3, n).
a(n) = A019538(n+3, n). (End)
MATHEMATICA
Table[n!*StirlingS2[n+3, n], {n, 0, 30}] (* G. C. Greubel, Jun 20 2022 *)
PROG
(PARI) a(n)=(n+3)!*n^2*(n+1)/48 \\ Charles R Greathouse IV, Nov 02 2011
(Magma) [Factorial(n+3)*n^2*(n+1)/48: n in [0..20]]; // Vincenzo Librandi, Nov 18 2011
(SageMath) [factorial(n)*stirling_number2(n+3, n) for n in (0..30)] # G. C. Greubel, Jun 20 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved