A037962 a(n) = n*(15*n^3 + 30*n^2 + 5*n - 2)*(n+4)!/5760.

0, 1, 62, 1806, 40824, 834120, 16435440, 322494480, 6411968640, 130456085760, 2731586457600, 59056027430400, 1320663933388800, 30575780537702400, 733062897120153600, 18198613875746304000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

Cf. A000142, A001298, A019538, A131689.

Programs

Magma

[Factorial(n)*StirlingSecond(n+4,n): n in [0..30]]; // G. C. Greubel, Jun 20 2022

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 *)

SageMath

[factorial(n)*stirling_number2(n+4,n) for n in (0..30)] # G. C. Greubel, Jun 20 2022

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)