OFFSET
0,5
COMMENTS
Matrix product of Stirling2 with unsigned Stirling1 triangle.
For the subtriangle without column no. m=0 and row no. n=0 see A079641.
The reversed matrix product |S1|. S2 is given in A271703.
As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
E.g.f. for row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
From Natalia L. Skirrow, Dec 22 2025: (Start)
Number of partitions of [n] into a set of k nonempty cycles of nonempty sets.
For d>0, d-th diagonal T(d+x,x) is a polynomial of the form x*(1+x)*(2+x)*...*(d+x)*P_d(x), with P_d of degree d-1. It appears that for d even, P_d is divisible by d-1+x again. (End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First ten rows and more.
FORMULA
E.g.f. of column k (with leading zeros): (f(x)^k)/k! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
Sum_{k=0..n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - Philippe Deléham, Nov 19 2011
EXAMPLE
Triangle begins:
1;
0, 1;
0, 2, 1;
0, 6, 6, 1;
0, 26, 36, 12, 1;
0, 150, 250, 120, 20, 1;
0, 1082, 2040, 1230, 300, 30, 1;
MAPLE
# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n, 1/2), 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
rows = 9;
t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* Peter Luschny, Jun 27 2019 *)
PROG
(SageMath)
def a_row(n):
s = sum(stirling_number2(n, k)*rising_factorial(x, k) for k in (0..n))
return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 04 2007
EXTENSIONS
New name by Peter Luschny, Jun 27 2019
STATUS
approved
