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A351804 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^n*x).

%I A351804 #19 May 13 2025 11:54:21
%S A351804 1,1,21,28800,6702928485,485036145970949475,
%T A351804 17284020213927891173772415260,
%U A351804 439885788765576174397949231373608504971360,10926401685584312222862714944076761452123218197332439365413,346792877099311752547903589477147000220953930332269111366383185472249165168535
%N A351804 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^n*x).
%H A351804 Alois P. Heinz, <a href="/A351804/b351804.txt">Table of n, a(n) for n = 0..26</a>
%F A351804 a(n) = Sum_{p in {1..n}^n : p_i <= p_{i+1}} Product_{j=1..n} p_j^n.
%F A351804 a(n) ~ c * n^(n^2), where c = 1/QPochhammer(exp(-1)) = 1.98244090741... - _Vaclav Kotesovec_, May 13 2025
%e A351804 a(2) = (1*1)^2 + (1*2)^2 + (2*2)^2 = 1 + 4 + 16 = 21.
%p A351804 b:= proc(n, k, p) option remember; `if`(k=0, 1,
%p A351804       add(b(j, k-1, p)*j^p, j=1..n))
%p A351804     end:
%p A351804 a:= n-> b(n$3):
%p A351804 seq(a(n), n=0..9);
%t A351804 Table[SeriesCoefficient[Product[1/(1 - k^n*x), {k, 1, n}], {x, 0, n}], {n, 0, 10}] (* _Vaclav Kotesovec_, May 13 2025 *)
%Y A351804 Cf. A036740, A351800.
%K A351804 nonn
%O A351804 0,3
%A A351804 _Alois P. Heinz_, Feb 19 2022