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A001754
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Lah numbers: n!*C(n-1,2)/6.
(Formerly M4863 N2079)
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5
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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; internal format)
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OFFSET
| 1,4
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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 (FrankTAW(AT)Netscape.net), Oct 25 2006
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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).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..300
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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). [From Milan R. Janjic (agnus(AT)blic.net), Mar 01 2009]
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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 (zerinvarylajos(AT)yahoo.com), Apr 29 2007
restart: 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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2009]
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PROG
| (MAGMA) [Factorial(n)*Binomial(n-1, 2)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011
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CROSSREFS
| Column 3 of A008297. Cf. A053495.
Column m=3 of unsigned triangle A111596.
Cf. A005990.
Sequence in context: A129329 A129332 A004291 * A037511 A037694 A067102
Adjacent sequences: A001751 A001752 A001753 * A001755 A001756 A001757
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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