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%I #24 Sep 08 2022 08:45:02
%S 1,1,2,1,6,3,1,12,24,4,1,20,90,80,5,1,30,240,540,240,6,1,42,525,2240,
%T 2835,672,7,1,56,1008,7000,17920,13608,1792,8,1,72,1764,18144,78750,
%U 129024,61236,4608,9,1,90,2880,41160,272160,787500,860160,262440,11520,10
%N Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
%H G. C. Greubel, <a href="/A059300/b059300.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>
%F T(n,k) = binomial(n+1,n-k+1)*(n-k+1)^k. - _R. J. Mathar_, Mar 14 2013
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 6, 3;
%e 1, 12, 24, 4;
%e 1, 20, 90, 80, 5;
%e 1, 30, 240, 540, 240, 6;
%e 1, 42, 525, 2240, 2835, 672, 7;
%e ...
%t t[n_, k_] := Binomial[n + 1, k]*(n - k + 1)^k; Flatten@Table[t[n, k], {n, 0, 9}, {k, 0, n}] (* _Arkadiusz Wesolowski_, Mar 23 2013 *)
%o (Magma) /* As triangle: */ [[Binomial(n+1,n-k+1)*(n-k+1)^k: k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 22 2015
%o (PARI) for(n=0, 25, for(k=0, n, print1(binomial(n+1,k)*(n-k+1)^k, ", "))) \\ _G. C. Greubel_, Jan 05 2017
%Y There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.
%K nonn,tabl
%O 0,3
%A _N. J. A. Sloane_, Jan 25 2001
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