A001754 - OEIS

%I M4863 N2079 #48 May 01 2024 09:00:55

%S 0,0,1,12,120,1200,12600,141120,1693440,21772800,299376000,4390848000,

%T 68497228800,1133317785600,19833061248000,366148823040000,

%U 7113748561920000,145120470663168000,3101950060425216000,69337707233034240000,1617879835437465600000

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

%C 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

%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.

%D John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A001754/b001754.txt">Table of n, a(n) for n = 1..300</a>

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

%F 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

%F a(n) = Sum_{k>=1} k * A260665(n,k). - _Alois P. Heinz_, Nov 14 2015

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

%F From _Amiram Eldar_, May 02 2022: (Start)

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

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

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

%t Table[(n-2)*(n-1)*n!/12, {n, 21}] (* _Arkadiusz Wesolowski_, Nov 26 2012 *)

%t With[{nn=30},CoefficientList[Series[(x/(1-x))^3/6,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 04 2017 *)

%o (Magma) [Factorial(n)*Binomial(n-1, 2)/6: n in [1..25]]; // _Vincenzo Librandi_, Oct 11 2011

%o (Sage) [factorial(n-1)*binomial(n,3)/2 for n in (1..30)] # _G. C. Greubel_, May 10 2021

%Y Column 3 of A008297.

%Y Column m=3 of unsigned triangle A111596.

%Y Cf. A005990, A053495, A260665.

%Y Cf. A001113, A001620, A091725, A099285.

%K nonn,easy

%O 1,4

%A _N. J. A. Sloane_