A001778 Lah numbers: a(n) = n!*binomial(n-1,5)/6!.

1, 42, 1176, 28224, 635040, 13970880, 307359360, 6849722880, 155831195520, 3636061228800, 87265469491200, 2157837063782400, 55024845126451200, 1447576694865100800, 39291367432052736000, 1100158288097476608000, 31767070568814637056000

Offset: 6

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
• John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 6..100

Crossrefs

Column 6 of A008297.

Column m=6 of unsigned triangle A111596.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[Factorial(n-6)*Binomial(n,6)*Binomial(n-1,5): n in [6..30]]; // G. C. Greubel, May 10 2021

Maple

A001778 := proc(n)
    n!*binomial(n-1,5)/6! ;
end proc:
seq(A001778(n),n=6..30) ; # R. J. Mathar, Jan 06 2021

Mathematica

With[{c=6!},Table[n!Binomial[n-1,5]/c,{n,6,24}]] (* Harvey P. Dale, May 25 2011 *)

Sage

[binomial(n,6)*factorial(n-1)/factorial(5) for n in range(6, 22)] # Zerinvary Lajos, Jul 07 2009

Formula

E.g.f.: ((x/(1-x))^6)/6!.

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,6,-6), (n>=6). - Milan Janjic, Mar 01 2009

D-finite with recurrence (-n+6)*a(n) +n*(n-1)*a(n-1)=0. - R. J. Mathar, Jan 06 2021

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=6} 1/a(n) = 570*(gamma - Ei(1)) + 1380*e - 2999, where gamma = A001620, Ei(1) = A091725 and e = A001113.

Sum_{n>=6} (-1)^n/a(n) = 15030*(gamma - Ei(-1)) - 9000/e - 8661, where Ei(-1) = -A099285. (End)

Extensions

More terms from Christian G. Bower, Dec 18 2001