A001754 Lah numbers: a(n) = n!*binomial(n-1,2)/6. (Formerly M4863 N2079)

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

Offset

1,4

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

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 anf e = A001113. (End)

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 *)

Prog

(Magma) [Factorial(n)*Binomial(n-1, 2)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011
(Sage) [factorial(n-1)*binomial(n, 3)/2 for n in (1..30)] # G. C. Greubel, May 10 2021

Crossrefs

Column 3 of A008297.

Column m=3 of unsigned triangle A111596.

Cf. A005990, A053495, A260665.

Cf. A001113, A001620, A091725, A099285.

Sequence in context: A266393 A129332 A004291 * A037511 A037694 A242810

Adjacent sequences: A001751 A001752 A001753 * A001755 A001756 A001757

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved