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A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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)

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)

# The function bell_matrix is defined in 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

There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248. A093375.

Sequence in context: A115597 A103371 A120257 * A214306 A156914 A289656

Adjacent sequences:  A059295 A059296 A059297 * A059299 A059300 A059301

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jan 25 2001

STATUS

approved

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Last modified November 17 10:07 EST 2018. Contains 317275 sequences. (Running on oeis4.)