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A172109 a(n) is the number of ordered partitions of {1,1,2,3,...,n-1}.

%I A172109 #37 Apr 15 2022 04:42:34
%S A172109 0,2,8,44,308,2612,25988,296564,3816548,54667412,862440068,
%T A172109 14857100084,277474957988,5584100659412,120462266974148,
%U A172109 2772968936479604,67843210855558628,1757952715142990612,48093560991292628228
%N A172109 a(n) is the number of ordered partitions of {1,1,2,3,...,n-1}.
%H A172109 G. C. Greubel, <a href="/A172109/b172109.txt">Table of n, a(n) for n = 1..400</a>
%H A172109 M. Griffiths and I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths/griffiths11.html">A generalization of Stirling Numbers of the Second Kind via a special multiset</a>, JIS 13 (2010) #10.2.5.
%F A172109 For n>=2, T_2(n) = Sum_{m=1..n} Sum_{l=0..m} C(m,l)*C(l+1,2)*(-1)^(m-l)*l^(n-2).
%F A172109 G.f.: 1/G(0) -1  where G(k) = 1 - x*(k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 23 2013
%F A172109 G.f.: 1/Q(0) -1, where Q(k) = 1 - x*(3*k+2) - 2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 03 2013
%F A172109 a(n) = Sum_{k=1..n-1} Stirling2(n-1,k)*(k+1)!. - _Karol A. Penson_, Sep 04 2015
%F A172109 a(n) ~ n! / (4 * log(2)^(n+1)). - _Vaclav Kotesovec_, Apr 15 2022
%t A172109 f[r_, n_]:= Sum[Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l, m}], {m, n}]; Join[{0}, Table[f[2, n], {n,2,30}]]
%o A172109 (Magma) [(&+[Factorial(j+1)*StirlingSecond(n-1,j): j in [1..n]]): n in [1..30]]; // _G. C. Greubel_, Apr 14 2022
%o A172109 (SageMath) [sum( factorial(j+1)*stirling_number2(n-1,j) for j in (1..n-1) ) for n in (1..30)] # _G. C. Greubel_, Apr 14 2022
%o A172109 (PARI) a(n) = sum(k=1, n-1, stirling(n-1,k,2)*(k+1)!); \\ _Michel Marcus_, Apr 14 2022
%Y A172109 Row sums of A172106.
%Y A172109 Cf. A005649. - _R. J. Mathar_, Jan 28 2010
%Y A172109 Cf. A083410.
%K A172109 nonn
%O A172109 1,2
%A A172109 _Martin Griffiths_, Jan 25 2010