A355576 Number A(n,k) of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= k^(i-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 7, 1, 0, 1, 1, 4, 24, 44, 1, 0, 1, 1, 5, 58, 541, 516, 1, 0, 1, 1, 6, 115, 3236, 35649, 11622, 1, 0, 1, 1, 7, 201, 12885, 713727, 6979689, 512022, 1, 0, 1, 1, 8, 322, 39656, 7173370, 627642640, 4085743032, 44588536, 1, 0

Offset

0,13

Links

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

Example

A(2,3) = 3: (1,1), (1,2), (1,3).
A(3,2) = 7: (1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,2,2), (1,2,3), (1,2,4).
A(3,3) = 24: (1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,1,5), (1,1,6), (1,1,7), (1,1,8), (1,1,9), (1,2,2), (1,2,3), (1,2,4), (1,2,5), (1,2,6), (1,2,7), (1,2,8), (1,2,9), (1,3,3), (1,3,4), (1,3,5), (1,3,6), (1,3,7), (1,3,8), (1,3,9).
Square array A(n,k) begins:
  1, 1,     1,       1,         1,           1,            1, ...
  1, 1,     1,       1,         1,           1,            1, ...
  0, 1,     2,       3,         4,           5,            6, ...
  0, 1,     7,      24,        58,         115,          201, ...
  0, 1,    44,     541,      3236,       12885,        39656, ...
  0, 1,   516,   35649,    713727,     7173370,     46769781, ...
  0, 1, 11622, 6979689, 627642640, 19940684251, 330736663032, ...

Maple

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

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, -Sum[A[j, k]*(-1)^(n-j)*Binomial[If[j==0, 1, k^j], n-j], {j, 0, n-1}]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Sep 21 2022, after Alois P. Heinz *)

Crossrefs

Columns k=1-9 give: A000012, A107354, A109055, A109056, A109057, A109058, A109059, A109060, A109061.

Rows n=1-4 give: A000012, A001477, A081436(k-1) for k>0, A354608.

Main diagonal gives A355561.

Sequence in context: A263857 A334895 A355262 * A198062 A347617 A226690

Adjacent sequences: A355573 A355574 A355575 * A355577 A355578 A355579

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 07 2022

Status

approved