A351339 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is Sum_{j=0..n} k^j * j^(n-j).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 18, 9, 0, 1, 5, 20, 48, 58, 23, 0, 1, 6, 30, 100, 201, 202, 66, 0, 1, 7, 42, 180, 516, 885, 762, 210, 0, 1, 8, 56, 294, 1105, 2756, 4116, 3114, 733, 0, 1, 9, 72, 448, 2094, 6955, 15300, 20298, 13754, 2781, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

G.f. of column k: Sum_{j>=0} (k*x)^j/(1 - j*x).

Example

Square array begins:
  1,  1,   1,    1,     1,     1,      1, ...
  0,  1,   2,    3,     4,     5,      6, ...
  0,  2,   6,   12,    20,    30,     42, ...
  0,  4,  18,   48,   100,   180,    294, ...
  0,  9,  58,  201,   516,  1105,   2094, ...
  0, 23, 202,  885,  2756,  6955,  15198, ...
  0, 66, 762, 4116, 15300, 45030, 112686, ...

Mathematica

T[0, k_] := 1; T[n_, 0] = 0; T[n_, k_] := Sum[k^j * j^(n - j), {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 08 2022 *)

Prog

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

Crossrefs

Columns k=0..3 give A000007, A026898(n-1), A351279, A351282.

Main diagonal gives A351340.

Sequence in context: A342120 A198793 A085388 * A361432 A294498 A292860

Adjacent sequences: A351336 A351337 A351338 * A351340 A351341 A351342

Keyword

nonn,tabl,easy

Author

Seiichi Manyama, Feb 08 2022

Status

approved