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).
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, 3740, 2880, 840, 1, 126, 2049, 11524, 30000, 39720, 26040, 6720, 1, 254, 6525, 54292, 207620, 418320, 460320, 262080, 60480, 1, 510, 20337, 243268, 1309560, 3755640, 6150480, 5779200, 2903040, 604800
Offset: 1
Examples
Triangle begins as: 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, 3740, 2880, 840; 1, 126, 2049, 11524, 30000, 39720, 26040, 6720; 1, 254, 6525, 54292, 207620, 418320, 460320, 262080, 60480; 1, 510, 20337, 243268, 1309560, 3755640, 6150480, 5779200, 2903040, 604800;
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
- M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
Programs
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Magma
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]]) >; [T(n,k,3): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 14 2022
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Mathematica
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}]]] -
SageMath
def T(n,k,m): if (n<3): return 0 else: return sum( (-1)^(k-j)*binomial(k,j)*binomial(j+m-1,m)*j^(n-m) for j in (1..k) ) flatten([[T(n,k,3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 14 2022
Formula
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.
Sum_{k=1..n} T_3(n, k) = A172110(n).
Sum_{k=1..n} (-1)^k*T_3(n, k) = 0. - G. C. Greubel, Apr 14 2022
Comments