A001754 Lah numbers: a(n) = n!*binomial(n-1,2)/6.

0, 0, 1, 12, 120, 1200, 12600, 141120, 1693440, 21772800, 299376000, 4390848000, 68497228800, 1133317785600, 19833061248000, 366148823040000, 7113748561920000, 145120470663168000, 3101950060425216000, 69337707233034240000, 1617879835437465600000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n+1) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0)^2, i.e., the sum of the squares of the positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters, Oct 25 2006

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..300

Crossrefs

Column 3 of A008297.

Column m=3 of unsigned triangle A111596.

Cf. A005990, A053495, A260665.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[Factorial(n)*Binomial(n-1, 2)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011

Maple

[seq(n!*binomial(n-1,2)/6, n=1..40)];

Mathematica

Table[(n-2)*(n-1)*n!/12, {n, 21}] (* Arkadiusz Wesolowski, Nov 26 2012 *)
With[{nn=30},CoefficientList[Series[(x/(1-x))^3/6,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2017 *)

Sage

[factorial(n-1)*binomial(n,3)/2 for n in (1..30)] # G. C. Greubel, May 10 2021

Formula

E.g.f.: ((x/(1-x))^3)/3!.

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,2,-4), n >= 2. - Milan Janjic, Mar 01 2009

a(n) = Sum_{k>=1} k * A260665(n,k). - Alois P. Heinz, Nov 14 2015

D-finite with recurrence (-n+5)*a(n) + (n-2)*(n-3)*a(n-1) = 0, n >= 4. - R. J. Mathar, Jan 06 2021

From Amiram Eldar, May 02 2022: (Start)

Sum_{n>=3} 1/a(n) = 6*(gamma - Ei(1)) + 9, where gamma = A001620 and Ei(1) = A091725.

Sum_{n>=3} (-1)^(n+1)/a(n) = 18*(gamma - Ei(-1)) - 12/e - 9, where Ei(-1) = -A099285 and e = A001113. (End)