OFFSET
0,2
COMMENTS
The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. - Peter Luschny, Mar 13 2009
T(n,k) is the sum of the products of the cardinality of the blocks (cells) in the set partitions of {1,2,..,n} into exactly k blocks.
From Peter Bala, Jul 22 2014: (Start)
Exponential Riordan array [(1+x)*exp(x), x*exp(x)].
Let M = A093375, the exponential Riordan array [(1+x)*exp(x), x], and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0 M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... - see the Example section. (End)
The Bell transform of n+1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Idempotent Number
EXAMPLE
Triangle begins
1;
2, 1;
3, 6, 1;
4, 24, 12, 1; ...
From Peter Bala, Jul 22 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1 \/1 \/1 \ /1 \
|2 1 ||0 1 ||0 1 | |2 1 |
|3 4 1 ||0 2 1 ||0 0 1 |... = |3 6 1 |
|4 9 6 1 ||0 3 4 1 ||0 0 2 1 | |4 24 12 1 |
|5 16 18 8 1||0 4 9 6 1||0 0 3 4 1| |5 80 90 20 1|
|... ||... ||... | |... | (End)
MAPLE
T:= (n, k)-> binomial(n+1, k+1)*(k+1)^(n-k): seq(seq(T(n, k), k=0..n), n=0..10); # Georg Fischer, Oct 27 2021
MATHEMATICA
t = Transpose[ Table[ Range[0, 11]! CoefficientList[ Series[(x Exp[x])^n/n!, {x, 0, 11}], x], {n, 11}]]; Table[ t[[n, k]], {n, 2, 11}, {k, n - 1}] // Flatten (* or simply *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Table[t[n, k], {n, 10}, {k, n}] // Flatten
PROG
(Magma) /* As triangle */ [[Binomial(n, k)*k^(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 22 2015
(Sage) # uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: n+1, 10) # Peter Luschny, Jan 18 2016
(PARI) for(n=1, 25, for(k=1, n, print1(binomial(n, k)*k^(n-k), ", "))) \\ G. C. Greubel, Jan 05 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jan 25 2001
STATUS
approved