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A001777 Lah numbers: a(n) = n! * binomial(n-1, 4)/5!.

%I A001777 M5213 N2267 #26 May 02 2022 02:59:10
%S A001777 1,30,630,11760,211680,3810240,69854400,1317254400,25686460800,
%T A001777 519437318400,10908183686400,237996734976000,5394592659456000,
%U A001777 126980411830272000,3101950060425216000,78582734864105472000,2062796790182768640000,56059536297908183040000
%N A001777 Lah numbers: a(n) = n! * binomial(n-1, 4)/5!.
%D A001777 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A001777 John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%D A001777 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001777 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001777 T. D. Noe, <a href="/A001777/b001777.txt">Table of n, a(n) for n = 5..100</a>
%F A001777 E.g.f.: ((x/(1-x))^5)/5!.
%F A001777 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,4,-6), (n>=4). - _Milan Janjic_, Mar 01 2009
%F A001777 From _Amiram Eldar_, May 02 2022: (Start)
%F A001777 Sum_{n>=5} 1/a(n) = 20*(Ei(1) - gamma) - 200*e + 1555/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
%F A001777 Sum_{n>=5} (-1)^(n+1)/a(n) = 1460*(gamma - Ei(-1)) - 880/e - 2515/3, where Ei(-1) = -A099285. (End)
%p A001777 A001777 := n-> n!*binomial(n-1,4)/5!;
%t A001777 Table[n! Binomial[n - 1, 4]/5!, {n, 5, 20}] (* _T. D. Noe_, Aug 10 2012 *)
%o A001777 (Sage) [binomial(n,5)*factorial (n-1)/factorial (4) for n in range(5, 21)] # _Zerinvary Lajos_, Jul 07 2009
%Y A001777 Column 5 of A008297. Cf. A053495.
%Y A001777 Column m=5 of unsigned triangle A111596.
%Y A001777 Cf. A001113, A001620, A091725, A099285.
%K A001777 nonn,easy
%O A001777 5,2
%A A001777 _N. J. A. Sloane_
%E A001777 More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu)