A368504 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * j^k.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 11, 21, 10, 1, 0, 1, 20, 60, 58, 15, 1, 0, 1, 37, 161, 244, 141, 21, 1, 0, 1, 70, 428, 900, 857, 318, 28, 1, 0, 1, 135, 1149, 3164, 4225, 2787, 685, 36, 1, 0, 1, 264, 3132, 10990, 18945, 18196, 8704, 1434, 45, 1

Offset: 0

Author

Seiichi Manyama, Dec 27 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  0,   0,    0,     0,      0,      0, ...
  1,  1,   1,    1,     1,      1,      1, ...
  1,  3,   6,   11,    20,     37,     70, ...
  1,  6,  21,   60,   161,    428,   1149, ...
  1, 10,  58,  244,   900,   3164,  10990, ...
  1, 15, 141,  857,  4225,  18945,  81565, ...
  1, 21, 318, 2787, 18196, 102501, 536046, ...
		

Links

• OEIS Wiki, Eulerian polynomials.

Crossrefs

Columns k=0..5 give A000012, A000217, A047520, A066999, A067534, A218376.

Main diagonal gives A368505.

Cf. A368486.

Programs

PARI

T(n, k) = sum(j=0, n, k^(n-j)*j^k);

Formula

G.f. of column k: x*A_k(x)/((1-k*x) * (1-x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.

T(0,k) = 0^k; T(n,k) = k*T(n-1,k) + n^k.