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A001754 Lah numbers: n!*C(n-1,2)/6.
(Formerly M4863 N2079)
11
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

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.

J. 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(sum(binomial(k,j) *stirling1(n,k) *stirling2(j,i) *x^(k-j), j=i..k), k=i..n) then a(n+1)=(-1)^n*f(n,2,-4), (n>=2). - Milan Janjic, Mar 01 2009

a(n) = Sum_{k>0} k * A260665(n,k). - Alois P. Heinz, Nov 14 2015

MAPLE

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

a:=n->sum((n-j)*n!/6, j=2..n): seq(a(n), n=1..21); # Zerinvary Lajos, Apr 29 2007

G(x):=x^3/(1-x)^3: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/3!, n=1..21); # Zerinvary Lajos, Apr 01 2009

MATHEMATICA

Table[(n - 2)*(n - 1)*n!/12, {n, 21}] (* Arkadiusz Wesolowski, Nov 26 2012 *)

PROG

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

CROSSREFS

Column 3 of A008297. Cf. A053495.

Column m=3 of unsigned triangle A111596.

Cf. A005990, A260665.

Sequence in context: A266393 A129332 A004291 * A037511 A037694 A242810

Adjacent sequences:  A001751 A001752 A001753 * A001755 A001756 A001757

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified June 22 12:17 EDT 2017. Contains 288613 sequences.