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A001755 Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.

%I A001755 M5096 N2207 #35 May 02 2022 02:58:56
%S A001755 1,20,300,4200,58800,846720,12700800,199584000,3293136000,57081024000,
%T A001755 1038874636800,19833061248000,396661224960000,8299373322240000,
%U A001755 181400588328960000,4135933413900288000,98228418580131840000,2426819753156198400000,62288373664342425600000
%N A001755 Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.
%D A001755 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A001755 John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%D A001755 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001755 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001755 T. D. Noe, <a href="/A001755/b001755.txt">Table of n, a(n) for n = 4..100</a>
%F A001755 E.g.f.: ((x/(1-x))^4)/4!.
%F A001755 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 A001755 D-finite with recurrence (-n+4)*a(n) +n*(n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 06 2021
%F A001755 From _Amiram Eldar_, May 02 2022: (Start)
%F A001755 Sum_{n>=4} 1/a(n) = 12*(Ei(1) - gamma + 2*e) - 80, where Ei(1) = A091725, gamma = A001620, and e = A001113.
%F A001755 Sum_{n>=4} (-1)^n/a(n) = 156*(gamma - Ei(-1)) - 96/e - 88, where Ei(-1) = -A099285. (End)
%p A001755 A001755 := n-> n!*binomial(n-1,3)/4!;
%t A001755 Table[n!Binomial[n-1, 3]/4!, {n, 4, 25}] (* _T. D. Noe_, Aug 10 2012 *)
%o A001755 (Sage) [binomial(n,4)*factorial (n-1)/6 for n in range(4, 21)] # _Zerinvary Lajos_, Jul 07 2009
%o A001755 (Magma) [Factorial(n-1)*Binomial(n, 4)/6: n in [4..30]]; // _G. C. Greubel_, May 10 2021
%Y A001755 Column 4 of A008297.
%Y A001755 Column m=4 of unsigned triangle A111596.
%Y A001755 Cf. A053495.
%Y A001755 Cf. A001113, A001620, A091725, A099285.
%K A001755 nonn,easy
%O A001755 4,2
%A A001755 _N. J. A. Sloane_
%E A001755 More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001