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

7,2

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

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)

Mathematica

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

Prog

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

Crossrefs

Column 7 of A008297 and unsigned A111596.

Column 6 of A001778.

Cf. A001113, A001620, A091725, A099285.

Sequence in context: A103726 A332859 A004387 * A111781 A221398 A280803

Adjacent sequences: A111594 A111595 A111596 * A111598 A111599 A111600

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 23 2005

Status

approved