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Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.
(Formerly M5096 N2207)
8

%I M5096 N2207 #35 May 02 2022 02:58:56

%S 1,20,300,4200,58800,846720,12700800,199584000,3293136000,57081024000,

%T 1038874636800,19833061248000,396661224960000,8299373322240000,

%U 181400588328960000,4135933413900288000,98228418580131840000,2426819753156198400000,62288373664342425600000

%N Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.

%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 T. D. Noe, <a href="/A001755/b001755.txt">Table of n, a(n) for n = 4..100</a>

%F E.g.f.: ((x/(1-x))^4)/4!.

%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)^n*f(n,4,-4), (n>=4). - _Milan Janjic_, Mar 01 2009

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

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

%F Sum_{n>=4} 1/a(n) = 12*(Ei(1) - gamma + 2*e) - 80, where Ei(1) = A091725, gamma = A001620, and e = A001113.

%F Sum_{n>=4} (-1)^n/a(n) = 156*(gamma - Ei(-1)) - 96/e - 88, where Ei(-1) = -A099285. (End)

%p A001755 := n-> n!*binomial(n-1,3)/4!;

%t Table[n!Binomial[n-1, 3]/4!, {n, 4, 25}] (* _T. D. Noe_, Aug 10 2012 *)

%o (Sage) [binomial(n,4)*factorial (n-1)/6 for n in range(4, 21)] # _Zerinvary Lajos_, Jul 07 2009

%o (Magma) [Factorial(n-1)*Binomial(n, 4)/6: n in [4..30]]; // _G. C. Greubel_, May 10 2021

%Y Column 4 of A008297.

%Y Column m=4 of unsigned triangle A111596.

%Y Cf. A053495.

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

%K nonn,easy

%O 4,2

%A _N. J. A. Sloane_

%E More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001