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).

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, 810, 210, 1, 78, 978, 4576, 10305, 12090, 7140, 1680, 1, 158, 3168, 22216, 74405, 134370, 134610, 70560, 15120, 1, 318, 9978, 101536, 483105, 1252650, 1882860, 1641360, 771120, 151200

Offset: 1

Author

Martin Griffiths, Jan 25 2010

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins as:
  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,     810,     210;
  1,  78,  978,   4576,  10305,   12090,    7140,    1680;
  1, 158, 3168,  22216,  74405,  134370,  134610,   70560,  15120;
  1, 318, 9978, 101536, 483105, 1252650, 1882860, 1641360, 771120, 151200;
		

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.

Crossrefs

This is related to A019538, A172106 and A172107.

Row sums give A172111.

Programs

Magma

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]]) >;
[T(n,k,4): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 14 2022

Mathematica

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}]]]

SageMath

def T(n,k,m):
    if (n<4): 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,4) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 14 2022

Formula

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.

Sum_{k=1.n} T_4(n, k) = A172111(n).

Sum_{k=1..n} (-1)^k*T_4(n, k) = 0. - G. C. Greubel, Apr 14 2022