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A332747 Number of compositions of n^2, such that each element of [n] is used at least once as a part.

%I A332747 #23 Sep 08 2021 11:13:04
%S A332747 1,1,3,72,6232,1621620,1241237520,2675188471920,15634073104902000,
%T A332747 239929277724680059440,9411787539302194544158080,
%U A332747 922671287397731617736789070720,221805878984619105095368813189002240,128660270226206951104782827202740054476800
%N A332747 Number of compositions of n^2, such that each element of [n] is used at least once as a part.
%C A332747 Some parts can be larger than n. Adding the condition that parts cannot be larger than n, we get A332721. Removing from A332721 the condition that each element of [n] has to be used, we get A332716.
%H A332747 Alois P. Heinz, <a href="/A332747/b332747.txt">Table of n, a(n) for n = 0..50</a>
%e A332747 a(4) = 6232: all permutations of 4321111111, 432211111, 43222111, 4322221, 43321111, 4332211, 433321, 4432111, 443221, 543211, 64321.
%p A332747 b:= proc(n, i, p, m) option remember; `if`(n=0, p!,
%p A332747       `if`(i<1, 0, (t-> add(b(n-i*j, i-1, p+j, t)/(j+
%p A332747       `if`(t=0, 1, 0))!, j=0..n/i))(`if`(i>m, m, 0))))
%p A332747     end:
%p A332747 a:= n-> b(n*(n-1)/2$2, n$2):
%p A332747 seq(a(n), n=0..15);
%t A332747 b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, p!,
%t A332747      If[i < 1, 0, Function[t, [b[n - i*j, i - 1, p + j, t]/(j +
%t A332747      If[t == 0, 1, 0])!, {j, 0, n/i}]][If[i > m, m, 0]]]];
%t A332747 a[n_] := b[n(n-1)/2, n(n-1)/2, n, n];
%t A332747 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Sep 08 2021, after _Alois P. Heinz_ *)
%Y A332747 Cf. A011782, A103488, A332716, A332721, A332796.
%K A332747 nonn
%O A332747 0,3
%A A332747 _Alois P. Heinz_, Feb 21 2020