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A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

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