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%I #22 Nov 12 2023 09:13:56
%S 1,1,0,1,2,0,1,6,3,0,1,12,24,4,0,1,20,90,80,5,0,1,30,240,540,240,6,0,
%T 1,42,525,2240,2835,672,7,0,1,56,1008,7000,17920,13608,1792,8,0,1,72,
%U 1764,18144,78750,129024,61236,4608,9,0,1,90,2880,41160
%N Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
%H G. C. Greubel, <a href="/A059299/b059299.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e Triangle begins:
%e 1,
%e 1, 0,
%e 1, 2, 0,
%e 1, 6, 3, 0,
%e 1, 12, 24, 4, 0,
%e 1, 20, 90, 80, 5, 0,
%e 1, 30, 240, 540, 240, 6, 0,
%e 1, 42, 525, 2240, 2835, 672, 7, 0,
%e ...
%p T := (n, k) -> binomial(n, k) * (n - k)^k:
%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%t t[n_, k_] := Binomial[n, k]*(n - k)^k; Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* _Arkadiusz Wesolowski_, Mar 23 2013 *)
%o (Magma) /* As triangle: */ [[Binomial(n,k)*(n-k)^k: k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 22 2015
%o (PARI) concat([1], for(n=0, 25, for(k=0, n, print1(binomial(n,k)*(n-k)^k, ", ")))) \\ _G. C. Greubel_, Jan 05 2017
%Y There are 4 versions: A059297-A059300.
%Y Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
%Y Row sums are A000248.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Jan 25 2001
%E Name corrected by _Peter Luschny_, Nov 12 2023
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