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A172110 a(n) is the number of ordered partitions of {1, 1, 1, 2, 3, ..., n-2}.

%I A172110 #14 Apr 15 2022 04:29:20
%S A172110 0,0,4,20,132,1076,10404,116180,1469892,20766836,323924964,5527326740,
%T A172110 102396386052,2046350191796,43876822764324,1004631156809300,
%U A172110 24463049576172612,631213045618035956,17203155473859536484
%N A172110 a(n) is the number of ordered partitions of {1, 1, 1, 2, 3, ..., n-2}.
%H A172110 G. C. Greubel, <a href="/A172110/b172110.txt">Table of n, a(n) for n = 1..420</a>
%H A172110 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 A172110 T_3(n) = Sum_{m=1..n} Sum_{j=0..m} binomial(m,j)*binomial(j+2,3)*(-1)^(m-j)*j^(n-3) for n > 2 with T_3(1) = T_3(2) = 0.
%F A172110 a(n) ~ n! / (12 * log(2)^(n+1)). - _Vaclav Kotesovec_, Apr 15 2022
%t A172110 f[r_, n_]:= f[r, n]= If[n<3, 0, Sum[Sum[Binomial[m, l]Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l, m}], {m, n}]]; Table[f[3, n], {n, 25}]
%o A172110 (Magma) [0,0] cat [(&+[ (&+[Binomial(k,j)*Binomial(j+2,3)*(-1)^(k-j)*j^(n-3): j in [0..k]]): k in [1..n]]): n in [3..25]]; // _G. C. Greubel_, Apr 15 2022
%o A172110 (Sage) [0,0]+[sum(sum(binomial(k,j)*binomial(j+2,3)*(-1)^(k+j)*j^(n-3) for j in (0..k)) for k in (1..n)) for n in (3..25)] # _G. C. Greubel_, Apr 15 2022
%Y A172110 Row sums of A172107.
%K A172110 nonn
%O A172110 1,3
%A A172110 _Martin Griffiths_, Jan 25 2010