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A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.

%I A339431 #14 Mar 14 2021 08:12:23
%S A339431 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,
%T A339431 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,
%U A339431 126,0,0,12,6,0,0,0,12,126,0,12,6,120,0,7
%N A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.
%H A339431 Alois P. Heinz, <a href="/A339431/b339431.txt">Table of n, a(n) for n = 0..10000</a>
%H A339431 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A339431 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A339431 a(55) = 120 because we have [25, 16, 9, 4, 1] (120 permutations).
%p A339431 b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!,
%p A339431      (s-> `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(i^2))
%p A339431     end:
%p A339431 a:= n-> b(n, 1, 0):
%p A339431 seq(a(n), n=0..100);  # _Alois P. Heinz_, Dec 04 2020
%t A339431 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]]]];
%t A339431 a[n_] := b[n, 1, 0];
%t A339431 a /@ Range[0, 100] (* _Jean-François Alcover_, Mar 14 2021, after _Alois P. Heinz_ *)
%Y A339431 Cf. A000290, A331844, A332304, A339367, A339419, A339430.
%K A339431 nonn,look
%O A339431 0,15
%A A339431 _Ilya Gutkovskiy_, Dec 04 2020