OFFSET
1,8
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.
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.
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
EXAMPLE
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;
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}]]]
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Martin Griffiths, Jan 25 2010
STATUS
approved