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A111597 Lah numbers: a(n) = n!*binomial(n-1,6)/7!.

%I A111597 #19 May 02 2022 02:59:26
%S A111597 1,56,2016,60480,1663200,43908480,1141620480,29682132480,779155977600,
%T A111597 20777492736000,565147802419200,15721384321843200,448059453172531200,
%U A111597 13097122477350912000,392913674320527360000,12101741169072242688000
%N A111597 Lah numbers: a(n) = n!*binomial(n-1,6)/7!.
%D A111597 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A111597 John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%H A111597 G. C. Greubel, <a href="/A111597/b111597.txt">Table of n, a(n) for n = 7..440</a>
%F A111597 E.g.f.: ((x/(1-x))^7)/7!.
%F A111597 a(n) = (n!/7!)*binomial(n-1, 7-1).
%F A111597 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,6,-8), (n>=6). - _Milan Janjic_, Mar 01 2009
%F A111597 From _Amiram Eldar_, May 02 2022: (Start)
%F A111597 Sum_{n>=7} 1/a(n) = 6342*(Ei(1) - gamma) - 8988*e + 80374/5, where Ei(1) = A091725, gamma = A001620, and e = A001113.
%F A111597 Sum_{n>=7} (-1)^(n+1)/a(n) = 170142*(gamma - Ei(-1)) - 101640/e - 490714/5, where Ei(-1) = -A099285. (End)
%t A111597 k = 7; a[n_] := n!*Binomial[n-1, k-1]/k!; Table[a[n], {n, k, 22}]  (* _Jean-François Alcover_, Jul 09 2013 *)
%o A111597 (Magma) [Factorial(n-7)*Binomial(n, 7)*Binomial(n-1, 6): n in [7..30]]; // _G. C. Greubel_, May 10 2021
%o A111597 (Sage) [factorial(n-7)*binomial(n, 7)*binomial(n-1, 6) for n in (7..30)] # _G. C. Greubel_, May 10 2021
%Y A111597 Column 7 of A008297 and unsigned A111596.
%Y A111597 Column 6 of A001778.
%Y A111597 Cf. A001113, A001620, A091725, A099285.
%K A111597 nonn,easy
%O A111597 7,2
%A A111597 _Wolfdieter Lang_, Aug 23 2005