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A001754 Lah numbers: a(n) = n!*binomial(n-1,2)/6.
(Formerly M4863 N2079)
15
0, 0, 1, 12, 120, 1200, 12600, 141120, 1693440, 21772800, 299376000, 4390848000, 68497228800, 1133317785600, 19833061248000, 366148823040000, 7113748561920000, 145120470663168000, 3101950060425216000, 69337707233034240000, 1617879835437465600000 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A266393 A129332 A004291 * A037511 A037694 A242810
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified February 21 03:13 EST 2024. Contains 370219 sequences. (Running on oeis4.)