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A111599 Lah numbers: a(n) = n!*binomial(n-1,8)/9!. 1

%I #21 May 02 2022 08:30:50

%S 1,90,4950,217800,8494200,309188880,10821610800,371026656000,

%T 12614906304000,428906814336000,14668613050291200,506733905373696000,

%U 17735686688079360000,630299019222512640000,22780807409042242560000

%N Lah numbers: a(n) = n!*binomial(n-1,8)/9!.

%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.

%D John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

%F E.g.f.: ((x/(1-x))^9)/9!.

%F a(n) = (n!/9!)*binomial(n-1, 9-1).

%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-1)*f(n,9,-9), n >= 9. - _Milan Janjic_, Mar 01 2009

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

%F Sum_{n>=9} 1/a(n) = 564552*(Ei(1) - gamma) - 264528*e - 873657/35, where Ei(1) = A091725, gamma = A001620, and e = A001113.

%F Sum_{n>=9} (-1)^(n+1)/a(n) = 28393416*(gamma - Ei(-1)) - 16938720/e - 573537159/35, where Ei(-1) = -A099285. (End)

%p part_ZL:=[S,{S=Set(U,card=r),U=Sequence(Z,card>=1)}, labeled]: seq(count(subs(r=9,part_ZL),size=m),m=9..23) ; # _Zerinvary Lajos_, Mar 09 2007

%t Table[n!*Binomial[n-1, 8]/9!, {n, 9, 30}] (* _Wesley Ivan Hurt_, Dec 10 2013 *)

%Y Column 9 of unsigned A008297 and A111596.

%Y Column 8: A111598.

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

%K nonn,easy

%O 9,2

%A _Wolfdieter Lang_, Aug 23 2005

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)