A111597 Lah numbers: a(n) = n!*binomial(n-1,6)/7!.

1, 56, 2016, 60480, 1663200, 43908480, 1141620480, 29682132480, 779155977600, 20777492736000, 565147802419200, 15721384321843200, 448059453172531200, 13097122477350912000, 392913674320527360000, 12101741169072242688000

Offset: 7

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 = 7..440

Crossrefs

Column 7 of A008297 and unsigned A111596.

Column 6 of A001778.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[Factorial(n-7)*Binomial(n, 7)*Binomial(n-1, 6): n in [7..30]]; // G. C. Greubel, May 10 2021

Mathematica

k = 7; a[n_] := n!*Binomial[n-1, k-1]/k!; Table[a[n], {n, k, 22}]  (* Jean-François Alcover, Jul 09 2013 *)

Sage

[factorial(n-7)*binomial(n, 7)*binomial(n-1, 6) for n in (7..30)] # G. C. Greubel, May 10 2021

Formula

E.g.f.: ((x/(1-x))^7)/7!.

a(n) = (n!/7!)*binomial(n-1, 7-1).

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) = (-1)^n*f(n,6,-8), (n>=6). - Milan Janjic, Mar 01 2009

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=7} 1/a(n) = 6342*(Ei(1) - gamma) - 8988*e + 80374/5, where Ei(1) = A091725, gamma = A001620, and e = A001113.

Sum_{n>=7} (-1)^(n+1)/a(n) = 170142*(gamma - Ei(-1)) - 101640/e - 490714/5, where Ei(-1) = -A099285. (End)