A303990 Triangle, read by rows: n^k * k^n, for n >= 1 and k = 1..n.

1, 2, 16, 3, 72, 729, 4, 256, 5184, 65536, 5, 800, 30375, 640000, 9765625, 6, 2304, 157464, 5308416, 121500000, 2176782336, 7, 6272, 750141, 39337984, 1313046875, 32934190464, 678223072849, 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656

Offset: 1

Author

Wolfdieter Lang, May 22 2018

Keywords

• nonn
• tabl
• easy

Comments

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.

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

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

The row sums give A303991.

Examples

			The triangle T(n, k) begins:
======================================================================
n\k |  1    2      3        4          5           6            7  ...
----+-----------------------------------------------------------------
1:  |  1
2:  |  2   16
3:  |  3   72    729
4:  |  4  256   5184    65536
5:  |  5  800  30375   640000    9765625
6:  |  6 2304 157464  5308416  121500000  2176782336
7:  |  7 6272 750141 39337984 1313046875 32934190464 678223072849
...
row n=8: 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656;
row n=9: 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121;
row n=10: 10, 102400, 59049000, 10485760000, 976562500000, 60466176000000, 2824752490000000, 107374182400000000, 3486784401000000000, 100000000000000000000;
...
		

Crossrefs

Cf. A062275, A303991.

Programs

Magma

/* As triangle */ [[n^k*k^n: k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, May 23 2018

Mathematica

Table[n^k k^n, {n, 10}, {k, n}] //Flatten (* Vincenzo Librandi, May 23 2018 *)

PARI

T(n, k) = n^k * k^n;
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 25 2018

Formula

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