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A172108 Triangle T_4(n, m), the number of surjective multi-valued functions from {1, 1, 1, 1, 2, 3, ..., n-3} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).

%I A172108 #18 May 30 2022 16:32:53
%S A172108 0,0,0,0,0,0,1,3,3,1,1,8,18,16,5,1,18,78,136,105,30,1,38,288,856,1205,
%T A172108 810,210,1,78,978,4576,10305,12090,7140,1680,1,158,3168,22216,74405,
%U A172108 134370,134610,70560,15120,1,318,9978,101536,483105,1252650,1882860,1641360,771120,151200
%N A172108 Triangle T_4(n, m), the number of surjective multi-valued functions from {1, 1, 1, 1, 2, 3, ..., n-3} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).
%C A172108 T_4(1, m) = T_4(2, m) = T_4(3, m) = 0 by definition. T_4(n, m) also gives the number of ordered partitions of {1, 1, 1, 1, 2, 3, ..., n-3} into exactly m parts.
%H A172108 G. C. Greubel, <a href="/A172108/b172108.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A172108 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 A172108 T_4(n, m) = Sum_{j=0..m} binomial(m,j)*binomial(j+3,4)*(-1)^(m-j)*j^(n-4), for n >= 4, with T(n, k) = 0 for n < 4.
%F A172108 Sum_{k=1.n} T_4(n, k) = A172111(n).
%F A172108 Sum_{k=1..n} (-1)^k*T_4(n, k) = 0. - _G. C. Greubel_, Apr 14 2022
%e A172108 Triangle begins as:
%e A172108   0;
%e A172108   0,   0;
%e A172108   0,   0,    0;
%e A172108   1,   3,    3,      1;
%e A172108   1,   8,   18,     16,      5;
%e A172108   1,  18,   78,    136,    105,      30;
%e A172108   1,  38,  288,    856,   1205,     810,     210;
%e A172108   1,  78,  978,   4576,  10305,   12090,    7140,    1680;
%e A172108   1, 158, 3168,  22216,  74405,  134370,  134610,   70560,  15120;
%e A172108   1, 318, 9978, 101536, 483105, 1252650, 1882860, 1641360, 771120, 151200;
%t A172108 f[r_, n_, m_]:= Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l,m}]; For[n = 4, n <= 10, n++, Print[Table[f[4, n, m], {m, 1, n}]]]
%o A172108 (Magma)
%o A172108 T:= func< n,k,m | n lt 4 select 0 else (&+[(-1)^(k+j)*Binomial(k,j)*Binomial(j+m-1,m)*j^(n-m): j in [1..k]]) >;
%o A172108 [T(n,k,4): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 14 2022
%o A172108 (SageMath)
%o A172108 def T(n,k,m):
%o A172108     if (n<4): return 0
%o A172108     else: return sum( (-1)^(k-j)*binomial(k,j)*binomial(j+m-1,m)*j^(n-m) for j in (1..k) )
%o A172108 flatten([[T(n,k,4) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 14 2022
%Y A172108 This is related to A019538, A172106 and A172107.
%Y A172108 Row sums give A172111.
%K A172108 nonn,tabl
%O A172108 1,8
%A A172108 _Martin Griffiths_, Jan 25 2010