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%I #20 Jul 08 2022 19:06:05
%S 1,1,2,24,3236,7173370,330736663032,382149784071841422,
%T 12983632019302863224103688,14912674110246473369128526689667934,
%U 654972005961623890774153743504185499487372010,1228018869478731662593970252736815943512232438560622483276
%N Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= n^(i-1).
%H Alois P. Heinz, <a href="/A355561/b355561.txt">Table of n, a(n) for n = 0..36</a>
%e a(0) = 1: ( ).
%e a(1) = 1: (1).
%e a(2) = 2: (1,1), (1,2).
%e a(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).
%p b:= proc(n, k, i) option remember; `if`(n=0, 1,
%p add(b(n-1, k, j), j=1..min(i, k^(n-1))))
%p end:
%p a:= n-> b(n$2, infinity):
%p seq(a(n), n=0..6);
%p # second Maple program:
%p b:= proc(n, k) option remember; `if`(n=0, 1, -add(
%p b(j, k)*(-1)^(n-j)*binomial(k^j, n-j), j=0..n-1))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..12);
%Y Main diagonal of A355576.
%Y Cf. A076113, A090588, A107354, A355519.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 06 2022
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