A037959 a(n) = n^2*(n+1)*(n+2)!/48.

6, 90, 1200, 15750, 211680, 2963520, 43545600, 673596000, 10977120000, 188367379200, 3399953356800, 64457449056000, 1281520880640000, 26676557107200000, 580481882652672000, 13183287756807168000

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

References

• Identity (1.19)/(n+3) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 3.

Links

• G. C. Greubel, Table of n, a(n) for n = 2..350

Crossrefs

Cf. A000142, A001297, A019538, A131689.

Cf. A037960, A037961, A037962, A037963.

Programs

Magma

[Factorial(n)*StirlingSecond(n+3,n)/(n+3): n in [2..30]]; // G. C. Greubel, Jun 20 2022

Mathematica

Table[(n+2)!n^2(n+1)/48,{n,2,20}] (* Harvey P. Dale, Jul 29 2021 *)

SageMath

[factorial(n)*stirling_number2(n+3, n)/(n+3) for n in (2..30)] # G. C. Greubel, Jun 20 2022

Formula

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

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

a(n) = (1/(n+3))*Sum_{j=0..n} (-1)^(n+j)*binomial(n,j)*j^(n+3).

a(n) = n!*StirlingS2(n+3, n)/(n+3).

a(n) = A037961(n)/(n+3).

a(n) = A131689(n+3, n).

a(n) = A019538(n+3, n).

E.g.f.: x*(1 + 6*x + 3*x^2)/(4*(1-x)^6). (End)