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A172107 Triangle T_3(n, m), the number of surjective multi-valued functions from {1, 1, 1, 2, 3, ..., n-2} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).

%I A172107 #16 Apr 15 2022 05:21:12
%S A172107 0,0,0,1,2,1,1,6,9,4,1,14,45,52,20,1,30,177,388,360,120,1,62,621,2260,
%T A172107 3740,2880,840,1,126,2049,11524,30000,39720,26040,6720,1,254,6525,
%U A172107 54292,207620,418320,460320,262080,60480,1,510,20337,243268,1309560,3755640,6150480,5779200,2903040,604800
%N A172107 Triangle T_3(n, m), the number of surjective multi-valued functions from {1, 1, 1, 2, 3, ..., n-2} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).
%C A172107 T_3(1, m) = T_3(2, m) = 0 by definition. T_3(n, m) also gives the number of ordered partitions of {1, 1, 1, 2, 3, ..., n-2} into exactly m parts.
%H A172107 G. C. Greubel, <a href="/A172107/b172107.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A172107 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 A172107 T_3(n, m) = Sum_{j=0..m} binomial(m, j)*binomial(j+2, 3)*(-1)^(m-j)*j^(n-3), for n >= 3, with T(1, 1) = T(2, 1) = T(2, 2) = 0.
%F A172107 Sum_{k=1..n} T_3(n, k) = A172110(n).
%F A172107 Sum_{k=1..n} (-1)^k*T_3(n, k) = 0. - _G. C. Greubel_, Apr 14 2022
%e A172107 Triangle begins as:
%e A172107   0;
%e A172107   0,   0;
%e A172107   1,   2,     1;
%e A172107   1,   6,     9,      4;
%e A172107   1,  14,    45,     52,      20;
%e A172107   1,  30,   177,    388,     360,     120;
%e A172107   1,  62,   621,   2260,    3740,    2880,     840;
%e A172107   1, 126,  2049,  11524,   30000,   39720,   26040,    6720;
%e A172107   1, 254,  6525,  54292,  207620,  418320,  460320,  262080,   60480;
%e A172107   1, 510, 20337, 243268, 1309560, 3755640, 6150480, 5779200, 2903040, 604800;
%t A172107 f[r_, n_, m_]:= Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l,m}]; For[n = 3, n <= 10, n++, Print[Table[f[3, n, m], {m, 1, n}]]]
%o A172107 (Magma)
%o A172107 T:= func< n,k,m | n lt 3 select 0 else (&+[(-1)^(k+j)*Binomial(k,j)*Binomial(j+m-1,m)*j^(n-m): j in [1..k]]) >;
%o A172107 [T(n,k,3): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 14 2022
%o A172107 (SageMath)
%o A172107 def T(n,k,m):
%o A172107     if (n<3): return 0
%o A172107     else: return sum( (-1)^(k-j)*binomial(k,j)*binomial(j+m-1,m)*j^(n-m) for j in (1..k) )
%o A172107 flatten([[T(n,k,3) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 14 2022
%Y A172107 This is related to A019538, A172106 and A172108.
%Y A172107 Row sums give A172110.
%K A172107 nonn,tabl
%O A172107 1,5
%A A172107 _Martin Griffiths_, Jan 25 2010