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

0,3

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

Cf. A000142, A001298, A019538, A131689.

Sequence in context: A248450 A115504 A296359 * A017778 A035726 A017725

Adjacent sequences: A037959 A037960 A037961 * A037963 A037964 A037965

Keyword

nonn

Author

N. J. A. Sloane

Status

approved