A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

1, 2, 1, 3, 6, 1, 4, 24, 12, 1, 5, 80, 90, 20, 1, 6, 240, 540, 240, 30, 1, 7, 672, 2835, 2240, 525, 42, 1, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 10, 11520, 262440, 860160, 787500, 272160, 41160

Offset: 0

Author

N. J. A. Sloane, Jan 25 2001

Keywords

• nonn
• tabl

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

Examples

			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)
		

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

Crossrefs

There are 4 versions: A059297, A059298, A059299, A059300.

Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.

Row sums are A000248. A093375.

Programs

Magma

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 22 2015

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

PARI

for(n=1, 25, for(k=1, n, print1(binomial(n,k)*k^(n-k), ", "))) \\ G. C. Greubel, Jan 05 2017

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