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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.
4
1, 1, 3, 72, 6232, 1621620, 1241237520, 2675188471920, 15634073104902000, 239929277724680059440, 9411787539302194544158080, 922671287397731617736789070720, 221805878984619105095368813189002240, 128660270226206951104782827202740054476800
OFFSET
0,3
COMMENTS
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.
LINKS
EXAMPLE
a(4) = 6232: all permutations of 4321111111, 432211111, 43222111, 4322221, 43321111, 4332211, 433321, 4432111, 443221, 543211, 64321.
MAPLE
b:= proc(n, i, p, m) option remember; `if`(n=0, p!,
`if`(i<1, 0, (t-> add(b(n-i*j, i-1, p+j, t)/(j+
`if`(t=0, 1, 0))!, j=0..n/i))(`if`(i>m, m, 0))))
end:
a:= n-> b(n*(n-1)/2$2, n$2):
seq(a(n), n=0..15);
MATHEMATICA
b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, p!,
If[i < 1, 0, Function[t, [b[n - i*j, i - 1, p + j, t]/(j +
If[t == 0, 1, 0])!, {j, 0, n/i}]][If[i > m, m, 0]]]];
a[n_] := b[n(n-1)/2, n(n-1)/2, n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 08 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 21 2020
STATUS
approved