A303990 - OEIS

%I #19 Sep 08 2022 08:46:21

%S 1,2,16,3,72,729,4,256,5184,65536,5,800,30375,640000,9765625,6,2304,

%T 157464,5308416,121500000,2176782336,7,6272,750141,39337984,

%U 1313046875,32934190464,678223072849,8,16384,3359232,268435456,12800000000,440301256704,12089663946752,281474976710656

%N Triangle, read by rows: n^k * k^n, for n >= 1 and k = 1..n.

%C Due to the symmetry of n^k * k^n under the exchange n <-> k, it is sufficient to consider n >= 1, and k = 1..n.

%C For the array n^k * k^n, with n >= 0 and k >= 0, read by antidiagonals, see the triangle A062275.

%C Thanks go to S. Heinemeyer for leading me to look at this.

%C The row sums give A303991.

%F T(n, k) = n^k * k^n, for n >= 1, k = 1..n.

%e The triangle T(n, k) begins:

%e ======================================================================

%e n\k |  1    2      3        4          5           6            7  ...

%e ----+-----------------------------------------------------------------

%e 1:  |  1

%e 2:  |  2   16

%e 3:  |  3   72    729

%e 4:  |  4  256   5184    65536

%e 5:  |  5  800  30375   640000    9765625

%e 6:  |  6 2304 157464  5308416  121500000  2176782336

%e 7:  |  7 6272 750141 39337984 1313046875 32934190464 678223072849

%e ...

%e row n=8: 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656;

%e row n=9: 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121;

%e row n=10: 10, 102400, 59049000, 10485760000, 976562500000, 60466176000000, 2824752490000000, 107374182400000000, 3486784401000000000, 100000000000000000000;

%e ...

%t Table[n^k k^n, {n, 10}, {k, n}] //Flatten (* _Vincenzo Librandi_, May 23 2018 *)

%o (Magma) /* As triangle */ [[n^k*k^n: k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, May 23 2018

%o (PARI) T(n, k) = n^k * k^n;

%o tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, May 25 2018

%Y Cf. A062275, A303991.

%K nonn,tabl,easy

%O 1,2

%A _Wolfdieter Lang_, May 22 2018