A037960 a(n) = n*(3*n+1)*(n+2)!/24.

0, 1, 14, 150, 1560, 16800, 191520, 2328480, 30240000, 419126400, 6187104000, 97037740800, 1612798387200, 28332944640000, 524813313024000, 10226013557760000, 209144207720448000, 4480594531725312000, 100357207837286400000, 2345925761384325120000, 57136703662028390400000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n>=1, a(n) is equal to the number of surjections from {1,2,..,n+2} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

References

• Identity (1.18) in H. W. Gould, Combinatorial Identities, Morgantown, 1972; page 3.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010 (identity 10.3, p.45)

Crossrefs

Cf. A000142, A001286, A001296, A019538, A037960, A131689.

Cf. A037959, A037961, A037962, A037963.

Programs

Magma

[Factorial(n+2)*n*(3*n+1)/24: n in [0..25]]; // Vincenzo Librandi, Feb 20 2017

Mathematica

Table[(n+2)!*n*(3n+1)/24,{n,0,20}] (* Harvey P. Dale, Oct 16 2014 *)

PARI

n*(3*n+1)*(n+2)!/24 \\ Charles R Greathouse IV, Nov 02 2011

SageMath

[factorial(n)*stirling_number2(n+2, n) for n in (0..30)] # G. C. Greubel, Jun 20 2022

Formula

a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Vladimir Kruchinin, Jun 01 2013

(3*n-2)*(n-1)*a(n) - n*(n+2)*(3*n+1)*a(n-1) = 0. - R. J. Mathar, Jul 26 2015

E.g.f.: x*(1 + 2*x)/(1 - x)^5. - Ilya Gutkovskiy, Feb 20 2017

From G. C. Greubel, Jun 20 2022: (Start)

a(n) = n!*StirlingS2(n+2, n).

a(n) = A131689(n+2, n).

a(n) = A019538(n+2, n). (End)

Extensions

More terms from Vincenzo Librandi, Feb 20 2017