A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.
0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 1, 6, 0, 0, 6, 6, 0, 0, 0, 0, 6, 1, 0, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 7, 6, 0, 0, 6, 6, 120, 6, 0, 0, 6, 0, 6, 12, 0, 1, 6, 126, 0, 0, 12, 6, 0, 0, 0, 12, 126, 0, 12, 6, 120, 0, 7
Offset: 0
Examples
a(55) = 120 because we have [25, 16, 9, 4, 1] (120 permutations).
Links
Programs
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Maple
b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!, (s-> `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(i^2)) end: a:= n-> b(n, 1, 0): seq(a(n), n=0..100); # Alois P. Heinz, Dec 04 2020 -
Mathematica
b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, With[{s = i^2}, If[s > n, 0, b[n, i + 1, p] + b[n - s, i + 1, p + 1]]]]; a[n_] := b[n, 1, 0]; a /@ Range[0, 100] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)