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%I #27 Sep 21 2022 10:39:52
%S 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,
%T 541,516,1,0,1,1,6,115,3236,35649,11622,1,0,1,1,7,201,12885,713727,
%U 6979689,512022,1,0,1,1,8,322,39656,7173370,627642640,4085743032,44588536,1,0
%N 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.
%H Alois P. Heinz, <a href="/A355576/b355576.txt">Antidiagonals n = 0..43, flattened</a>
%e A(2,3) = 3: (1,1), (1,2), (1,3).
%e 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).
%e 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).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, ...
%e 0, 1, 7, 24, 58, 115, 201, ...
%e 0, 1, 44, 541, 3236, 12885, 39656, ...
%e 0, 1, 516, 35649, 713727, 7173370, 46769781, ...
%e 0, 1, 11622, 6979689, 627642640, 19940684251, 330736663032, ...
%p A:= proc(n, k) option remember; `if`(n=0, 1, -add(
%p A(j, k)*(-1)^(n-j)*binomial(k^j, n-j), j=0..n-1))
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t 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}]];
%t Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Sep 21 2022, after _Alois P. Heinz_ *)
%Y Columns k=1-9 give: A000012, A107354, A109055, A109056, A109057, A109058, A109059, A109060, A109061.
%Y Rows n=1-4 give: A000012, A001477, A081436(k-1) for k>0, A354608.
%Y Main diagonal gives A355561.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Jul 07 2022