A346061 A(n,k) = n! * [x^n] (Sum_{j=0..n} k^(j*(j+1)/2) * x^j/j!)^(1/k) if k>0, A(n,0) = 0^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 23, 1, 0, 1, 1, 13, 199, 393, 1, 0, 1, 1, 21, 901, 17713, 13729, 1, 0, 1, 1, 31, 2861, 249337, 4572529, 943227, 1, 0, 1, 1, 43, 7291, 1900521, 264273961, 3426693463, 126433847, 1, 0

Offset

0,13

Comments

A(n,k) is odd if k >= 1 or n = 0.

Links

Alois P. Heinz, Antidiagonals n = 0..43, flattened

Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021.

Formula

E.g.f. of column k>0: (Sum_{j>=0} k^(j*(j+1)/2) * x^j/j!)^(1/k).

E.g.f. of column k=0: 1.

A(n,k) == 1 (mod k*(k-1)) for k >= 2 (see "general conjecture" in A178319 and link to proof by Richard Stanley above).

Example

Square array A(n,k) begins:
  1, 1,     1,       1,         1,          1, ...
  0, 1,     1,       1,         1,          1, ...
  0, 1,     3,       7,        13,         21, ...
  0, 1,    23,     199,       901,       2861, ...
  0, 1,   393,   17713,    249337,    1900521, ...
  0, 1, 13729, 4572529, 264273961, 6062674201, ...
  ...

Maple

A:= (n, k)-> `if`(k>0, n!*coeff(series(add(k^(j*(j+1)/2)*
             x^j/j!, j=0..n)^(1/k), x, n+1), x, n), k^n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Crossrefs

Columns k=0-3 give: A000007, A000012, A178315, A178319.

Rows n=0-2 give: A000012, A057427, A002061 (for k>0).

Main diagonal gives A342578.

Cf. A000217, A023813.

Sequence in context: A327622 A183134 A328747 * A053382 A031253 A291624

Adjacent sequences: A346058 A346059 A346060 * A346062 A346063 A346064

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 03 2021

Status

approved