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A001778 Lah numbers: a(n) = n!*binomial(n-1,5)/6!.

%I A001778 M5279 N2297 #34 May 02 2022 08:32:23
%S A001778 1,42,1176,28224,635040,13970880,307359360,6849722880,155831195520,
%T A001778 3636061228800,87265469491200,2157837063782400,55024845126451200,
%U A001778 1447576694865100800,39291367432052736000,1100158288097476608000,31767070568814637056000
%N A001778 Lah numbers: a(n) = n!*binomial(n-1,5)/6!.
%D A001778 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A001778 John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%D A001778 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001778 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001778 T. D. Noe, <a href="/A001778/b001778.txt">Table of n, a(n) for n = 6..100</a>
%F A001778 E.g.f.: ((x/(1-x))^6)/6!.
%F A001778 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,6,-6), (n>=6). - _Milan Janjic_, Mar 01 2009
%F A001778 D-finite with recurrence (-n+6)*a(n) +n*(n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 06 2021
%F A001778 From _Amiram Eldar_, May 02 2022: (Start)
%F A001778 Sum_{n>=6} 1/a(n) = 570*(gamma - Ei(1)) + 1380*e - 2999, where gamma = A001620, Ei(1) = A091725 and e = A001113.
%F A001778 Sum_{n>=6} (-1)^n/a(n) = 15030*(gamma - Ei(-1)) - 9000/e - 8661, where Ei(-1) = -A099285. (End)
%p A001778 A001778 := proc(n)
%p A001778     n!*binomial(n-1,5)/6! ;
%p A001778 end proc:
%p A001778 seq(A001778(n),n=6..30) ; # _R. J. Mathar_, Jan 06 2021
%t A001778 With[{c=6!},Table[n!Binomial[n-1,5]/c,{n,6,24}]] (* _Harvey P. Dale_, May 25 2011 *)
%o A001778 (Sage) [binomial(n,6)*factorial(n-1)/factorial(5) for n in range(6, 22)] # _Zerinvary Lajos_, Jul 07 2009
%o A001778 (Magma) [Factorial(n-6)*Binomial(n,6)*Binomial(n-1,5): n in [6..30]]; // _G. C. Greubel_, May 10 2021
%Y A001778 Column 6 of A008297.
%Y A001778 Column m=6 of unsigned triangle A111596.
%Y A001778 Cf. A001113, A001620, A091725, A099285.
%K A001778 nonn,easy
%O A001778 6,2
%A A001778 _N. J. A. Sloane_
%E A001778 More terms from _Christian G. Bower_, Dec 18 2001