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A059300 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.

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