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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.
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%I #12 Jul 10 2022 08:35:49

%S 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,

%T 32,1176,9280,5235,132,1,1,1,64,7280,249776,804322,102756,429,1,1,1,

%U 128,44640,6518784,119088660,109506040,2464898,1430,1

%N 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.

%H Alois P. Heinz, <a href="/A355614/b355614.txt">Antidiagonals n = 0..60, flattened</a>

%e A(2,2) = 4: (1,1), (1,2), (1,3), (1,4).

%e A(2,3) = 8: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8).

%e A(3,1) = 5: (1,1,1), (1,1,2), (1,1,3), (1,2,2), (1,2,3).

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 4, 8, 16, 32, ...

%e 1, 5, 30, 188, 1176, 7280, ...

%e 1, 14, 340, 9280, 249776, 6518784, ...

%e 1, 42, 5235, 804322, 119088660, 16633660072, ...

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(

%p A(j-1, k)*(-1)^(n-j)*binomial(j^k, n-j+1), j=1..n))

%p end:

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

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

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

%Y Main diagonal gives A355613.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Jul 09 2022