A111599 Lah numbers: a(n) = n!*binomial(n-1,8)/9!.

1, 90, 4950, 217800, 8494200, 309188880, 10821610800, 371026656000, 12614906304000, 428906814336000, 14668613050291200, 506733905373696000, 17735686688079360000, 630299019222512640000, 22780807409042242560000

Offset: 9

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.

Crossrefs

Column 9 of unsigned A008297 and A111596.

Column 8: A111598.

Cf. A001113, A001620, A091725, A099285.

Programs

Maple

part_ZL:=[S,{S=Set(U,card=r),U=Sequence(Z,card>=1)}, labeled]: seq(count(subs(r=9,part_ZL),size=m),m=9..23) ; # Zerinvary Lajos, Mar 09 2007

Mathematica

Table[n!*Binomial[n-1, 8]/9!, {n, 9, 30}] (* Wesley Ivan Hurt, Dec 10 2013 *)

Formula

E.g.f.: ((x/(1-x))^9)/9!.

a(n) = (n!/9!)*binomial(n-1, 9-1).

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-1)*f(n,9,-9), n >= 9. - Milan Janjic, Mar 01 2009

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=9} 1/a(n) = 564552*(Ei(1) - gamma) - 264528*e - 873657/35, where Ei(1) = A091725, gamma = A001620, and e = A001113.

Sum_{n>=9} (-1)^(n+1)/a(n) = 28393416*(gamma - Ei(-1)) - 16938720/e - 573537159/35, where Ei(-1) = -A099285. (End)