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

%I A059298 #49 Feb 16 2025 08:32:43
%S A059298 1,2,1,3,6,1,4,24,12,1,5,80,90,20,1,6,240,540,240,30,1,7,672,2835,
%T A059298 2240,525,42,1,8,1792,13608,17920,7000,1008,56,1,9,4608,61236,129024,
%U A059298 78750,18144,1764,72,1,10,11520,262440,860160,787500,272160,41160
%N A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.
%C A059298 The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. - _Peter Luschny_, Mar 13 2009
%C A059298 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.
%C A059298 From _Peter Bala_, Jul 22 2014: (Start)
%C A059298 Exponential Riordan array [(1+x)*exp(x), x*exp(x)].
%C A059298 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
%C A059298   /I_k 0\
%C A059298   \ 0  M/
%C A059298 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)
%C A059298 The Bell transform of n+1. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%D A059298 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
%H A059298 G. C. Greubel, <a href="/A059298/b059298.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059298 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>
%e A059298 Triangle begins
%e A059298 1;
%e A059298 2, 1;
%e A059298 3, 6, 1;
%e A059298 4, 24, 12, 1; ...
%e A059298 From _Peter Bala_, Jul 22 2014: (Start)
%e A059298 With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
%e A059298 /1          \/1        \/1        \      /1           \
%e A059298 |2  1       ||0 1      ||0 1      |      |2  1        |
%e A059298 |3  4  1    ||0 2 1    ||0 0 1    |... = |3  6  1     |
%e A059298 |4  9  6 1  ||0 3 4 1  ||0 0 2 1  |      |4 24 12  1  |
%e A059298 |5 16 18 8 1||0 4 9 6 1||0 0 3 4 1|      |5 80 90 20 1|
%e A059298 |...        ||...      ||...      |      |...         | (End)
%p A059298 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
%t A059298 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 A059298 t[n_, k_] := Binomial[n, k]*k^(n - k); Table[t[n, k], {n, 10}, {k, n}] // Flatten
%o A059298 (Magma) /* As triangle */ [[Binomial(n,k)*k^(n-k): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Aug 22 2015
%o A059298 (Sage) # uses[bell_matrix from A264428]
%o A059298 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A059298 bell_matrix(lambda n: n+1, 10) # _Peter Luschny_, Jan 18 2016
%o A059298 (PARI) for(n=1, 25, for(k=1, n, print1(binomial(n,k)*k^(n-k), ", "))) \\ _G. C. Greubel_, Jan 05 2017
%Y A059298 There are 4 versions: A059297, A059298, A059299, A059300.
%Y A059298 Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
%Y A059298 Row sums are A000248. A093375.
%K A059298 nonn,tabl
%O A059298 0,2
%A A059298 _N. J. A. Sloane_, Jan 25 2001