A156353 -id:A156353 - cp's OEIS frontend

A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

2, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073

Offset: 0

Henry Bottomley, Jun 08 2000

Columns and rows are A000012 (apart from first term), A000027, A001580, A001585, A001589, A001593 etc. Diagonals include A013499 (apart from first two terms), A051442, A051489.
Cf. A055651.
Contribution from Franklin T. Adams-Watters, Oct 26 2009: (Start)
Rows/columns: A000027, A001580, A001585, A001589, A001593, A001594, A001596.
Main diagonal is 2 * A000312. More diagonals: A051442, A051489, A155539.
Cf. A076980, A156353, A156354. (End)

E.g.f. Sum(n,m, T(n,m)/(n! m!)) = e^(x e^y) + e^(y e^x). [From Franklin T. Adams-Watters, Oct 26 2009]

A220417 Table T(n,k) = k^n - n^k, n, k > 0, read by descending antidiagonals.

Original entry on oeis.org

0, 1, -1, 2, 0, -2, 3, 1, -1, -3, 4, 0, 0, 0, -4, 5, -7, -17, 17, 7, -5, 6, -28, -118, 0, 118, 28, -6, 7, -79, -513, -399, 399, 513, 79, -7, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, 9, -431, -6049, -13983, -7849, 7849, 13983, 6049, 431, -9, 10, -924, -18954, -61440, -61318, 0, 61318, 61440, 18954, 924, -10

Offset: 1

Boris Putievskiy, Dec 14 2012

			The table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
   0   1     2      3       4       5 ...
  -1   0     1      0      -7     -28 ...
  -2  -1     0    -17    -118    -513 ...
  -3   0    17      0    -399   -2800 ...
  -4   7   118    399       0   -7849 ...
  -5  28   513   2800    7849       0 ...
  ...
The start of the sequence as a triangular array, read by rows (i.e., descending antidiagonals of T(n,k)), is as follows:
  0;
  1,  -1;
  2,   0,   -2;
  3,   1,   -1, -3;
  4,   0,    0,  0,  -4;
  5,  -7,  -17, 17,   7, -5;
  6, -28, -118,  0, 118, 28, -6;
  ...
In the above triangle, row number m contains m numbers: m^1 - 1^m, (m-1)^2 - 2^(m-1), ..., 1^m - m^1.

Boris Putievskiy, Rows n = 1..76 of triangle, flattened

Cf. A002260, A004736, A051128, A051129, A055651, A156353.

matrix(9, 9, n, k, k^n - n^k) \\ Michel Marcus, Oct 04 2019

t=int((math.sqrt(8*n-7) - 1)/ 2)
m=((t*t+3*t+4)/2-n)**(n-t*(t+1)/2)-(n-t*(t+1)/2)**((t*t+3*t+4)/2-n)

As a linear array, the sequence is a(n) = A004736(n)^A002260(n) - A002260(n)^A004736(n) or

a(n) = ((t*t + 3*t + 4)/2 - n)^(n - t*(t + 1)/2) - (n - t*(t + 1)/2)^((t*t + 3*t + 4)/2 - n) where t = floor((-1 + sqrt(8*n - 7))/2).

cp's OEIS Frontend

A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

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A220417 Table T(n,k) = k^n - n^k, n, k > 0, read by descending antidiagonals.

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