A001755 Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.

1, 20, 300, 4200, 58800, 846720, 12700800, 199584000, 3293136000, 57081024000, 1038874636800, 19833061248000, 396661224960000, 8299373322240000, 181400588328960000, 4135933413900288000, 98228418580131840000, 2426819753156198400000, 62288373664342425600000

Offset: 4

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 = 4..100

Crossrefs

Column 4 of A008297.

Column m=4 of unsigned triangle A111596.

Cf. A053495.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[Factorial(n-1)*Binomial(n, 4)/6: n in [4..30]]; // G. C. Greubel, May 10 2021

Maple

A001755 := n-> n!*binomial(n-1,3)/4!;

Mathematica

Table[n!Binomial[n-1, 3]/4!, {n, 4, 25}] (* T. D. Noe, Aug 10 2012 *)

Sage

[binomial(n,4)*factorial (n-1)/6 for n in range(4, 21)] # Zerinvary Lajos, Jul 07 2009

Formula

E.g.f.: ((x/(1-x))^4)/4!.

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,4,-4), (n>=4). - Milan Janjic, Mar 01 2009

D-finite with recurrence (-n+4)*a(n) +n*(n-1)*a(n-1)=0. - R. J. Mathar, Jan 06 2021

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=4} 1/a(n) = 12*(Ei(1) - gamma + 2*e) - 80, where Ei(1) = A091725, gamma = A001620, and e = A001113.

Sum_{n>=4} (-1)^n/a(n) = 156*(gamma - Ei(-1)) - 96/e - 88, where Ei(-1) = -A099285. (End)

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001