A037961 a(n) = n^2*(n+1)*(n+3)!/48.

0, 1, 30, 540, 8400, 126000, 1905120, 29635200, 479001600, 8083152000, 142702560000, 2637143308800, 50999300352000, 1031319184896000, 21785854970880000, 480178027929600000, 11029155770400768000

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

Cf. A000142, A001286, A001297, A019538, A037960, A131689.

Sequence in context: A270499 A286975 A139626 * A143399 A293103 A075510

Adjacent sequences: A037958 A037959 A037960 * A037962 A037963 A037964

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved