A355614 Number A(n,k) of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= i^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 5, 1, 1, 1, 8, 30, 14, 1, 1, 1, 16, 188, 340, 42, 1, 1, 1, 32, 1176, 9280, 5235, 132, 1, 1, 1, 64, 7280, 249776, 804322, 102756, 429, 1, 1, 1, 128, 44640, 6518784, 119088660, 109506040, 2464898, 1430, 1

Offset

0,9

Links

Alois P. Heinz, Antidiagonals n = 0..60, flattened

Example

A(2,2) = 4: (1,1), (1,2), (1,3), (1,4).
A(2,3) = 8: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8).
A(3,1) = 5: (1,1,1), (1,1,2), (1,1,3), (1,2,2), (1,2,3).
Square array A(n,k) begins:
  1,  1,    1,      1,         1,           1, ...
  1,  1,    1,      1,         1,           1, ...
  1,  2,    4,      8,        16,          32, ...
  1,  5,   30,    188,      1176,        7280, ...
  1, 14,  340,   9280,    249776,     6518784, ...
  1, 42, 5235, 804322, 119088660, 16633660072, ...

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(j-1, k)*(-1)^(n-j)*binomial(j^k, n-j+1), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Crossrefs

Columns k=0-2 give: A000012, A000108, A209440.

Rows n=1-2 give: A000012, A000079.

Main diagonal gives A355613.

Sequence in context: A291118 A245184 A284414 * A140274 A095231 A303697

Adjacent sequences: A355611 A355612 A355613 * A355615 A355616 A355617

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 09 2022

Status

approved