A016804 -id:A016804 - cp's OEIS frontend

A155955 Triangle read by rows: T(n,k) = (k*n)^k, 0 <= k <= n.

1, 1, 1, 1, 2, 16, 1, 3, 36, 729, 1, 4, 64, 1728, 65536, 1, 5, 100, 3375, 160000, 9765625, 1, 6, 144, 5832, 331776, 24300000, 2176782336, 1, 7, 196, 9261, 614656, 52521875, 5489031744, 678223072849, 1, 8, 256, 13824, 1048576, 102400000, 12230590464

Offset: 0

Reinhard Zumkeller, Jan 31 2009

T(n,0)  = 1;
T(n,1)  = n for n > 0;
T(n,2)  = A016742(n) for n > 1;
T(n,3)  = A016767(n) for n > 2;
T(n,4)  = A016804(n) for n > 3;
T(n,5)  = A016853(n) for n > 4;
T(n,6)  = A016914(n) for n > 5;
T(n,7)  = A016987(n) for n > 6;
T(n,8)  = A017072(n) for n > 7;
T(n,9)  = A017169(n) for n > 8;
T(n,10) = A017278(n) for n > 9;
T(n,11) = A017399(n) for n > 10;
T(n,12) = A017532(n) for n > 11;
T(n,n)  = A062206(n).

			Triangle begins:
  1;
  1, 1;
  1, 2,  16;
  1, 3,  36,  729;
  1, 4,  64, 1728,  65536;
  1, 5, 100, 3375, 160000,  9765625;
  1, 6, 144, 5832, 331776, 24300000, 2176782336;
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A000312.

[[(n*k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Sep 15 2018

Table[If[n == 0, 1, If[ k == 0, 1, (k*n)^k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 15 2018 *)

for(n=0,10, for(k=0,n, print1((k*n)^k, ", "))) \\ G. C. Greubel, Sep 15 2018

A229213 Square array of denominators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 16, 27, 4, 36, 216, 256, 5, 64, 729, 4096, 3125, 6, 100, 1728, 20736, 100000, 46656, 7, 144, 3375, 65536, 759375, 2985984, 823543, 8, 196, 5832, 160000, 3200000, 34012224, 105413504, 16777216, 9, 256

Offset: 1

Jean-François Alcover, Sep 16 2013

Limit(t(n,k), n -> infinity) = exp(1/k).
1st row = A000027
2nd row = A016742
3rd row = A016767
4th row = A016804
5th row = A016853
1st column = A000312
2nd column = A062971
3rd column = A091482
4th column = A091483

			Table of fractions begins:
   2,       3/2,        4/3,         5/4, ...
  9/4,     25/16,      49/36,       81/64, ...
64/27,   343/216,   1000/729,    2197/1728, ...
625/256, 6561/4096, 28561/20736, 83521/65536, ...
...
Table of denominators begins:
1,      2,     3,     4, ...
4,     16,    36,    64, ...
27,   216,   729,  1728, ...
256, 4096, 20736, 65536, ...
...
Triangle of antidiagonals begins:
1;
2, 4;
3, 16, 27;
4, 36, 216, 256;
...

Cf. A229212(numerators), A016742, A016767, A016804, A016853, A000312, A062971, A091482, A091483.

t[n_, k_] := (1+1/(k*n))^n; Table[t[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten // Denominator

cp's OEIS Frontend

A155955 Triangle read by rows: T(n,k) = (k*n)^k, 0 <= k <= n.

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