A111598 Lah numbers: a(n) = n!*binomial(n-1,7)/8!.

1, 72, 3240, 118800, 3920400, 122316480, 3710266560, 111307996800, 3339239904000, 100919250432000, 3088129063219200, 96012739965542400, 3040403432242176000, 98228418580131840000, 3241537813144350720000

Offset: 8

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• nonn
• easy

References

• Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
• John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

Links

• G. C. Greubel, Table of n, a(n) for n = 8..440

Crossrefs

Column 8 of unsigned A008297 and A111596.

Column 7 of A111597.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[Factorial(n-8)*Binomial(n,8)*Binomial(n-1,7): n in [8..35]]; // G. C. Greubel, May 10 2021

Mathematica

Table[(n-8)!*Binomial[n-1,7]*Binomial[n,8], {n,8,35}] (* G. C. Greubel, May 10 2021 *)

Sage

[factorial(n-8)*binomial(n,8)*binomial(n-1,7) for n in (8..35)] # G. C. Greubel, May 10 2021

Formula

E.g.f.: ((x/(1-x))^8)/8!.

a(n) = (n!/8!)*binomial(n-1, 8-1).

If we define f(n,i,x) = Sum_{k=i..n}(Sum_{j=i..k} (binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) ) ) then a(n) = (-1)^n*f(n,8,-8), (n>=8). - Milan Janjic, Mar 01 2009

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=8} 1/a(n) = 61096*(gamma - Ei(1)) + 54544*e - 338732/5, where gamma = A001620, Ei(1) = A091725 and e = A001113.

Sum_{n>=8} (-1)^n/a(n) = 2107448*(gamma - Ei(-1)) - 1257760/e - 6080436/5, where Ei(-1) = -A099285. (End)