A037963 a(n) = n^2*(n+1)*(3*n^2 + 7*n - 2)*(n+5)!/11520.

0, 1, 126, 5796, 186480, 5103000, 129230640, 3162075840, 76592355840, 1863435974400, 45950224320000, 1155068769254400, 29708792431718400, 783699448602470400, 21234672840116736000, 591499300737945600000

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+5} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

References

• Identity (1.21) 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, A019538, A112494, A131689.

Programs

Magma

[Factorial(n)*StirlingSecond(n+5,n): n in [0..30]]; // G. C. Greubel, Jun 20 2022

Mathematica

Table[n!*StirlingS2[n+5, n], {n,0,30}] (* G. C. Greubel, Jun 20 2022 *)

SageMath

[factorial(n)*stirling_number2(n+5,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+5).

a(n) = n!*StirlingS2(n+5, n).

a(n) = A131689(n+5, n).

a(n) = A019538(n+5, n).

E.g.f.: x*(1 + 52*x + 328*x^2 + 444*x^3 + 120*x^4)/(1-x)^11. (End)