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A300715 Number of compositions (ordered partitions) of n into squares that do not divide n.

%I A300715 #10 Mar 15 2018 19:23:57
%S A300715 1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,3,0,0,0,4,3,0,0,7,6,0,0,14,10,4,0,
%T A300715 22,20,10,0,32,39,20,0,49,70,42,0,12,116,88,0,128,156,174,11,207,3,
%U A300715 320,0,333,551,575,0,555,914,0,0,959,1502,1829,44,1691,2486,3192,0,3000,4172,4005
%N A300715 Number of compositions (ordered partitions) of n into squares that do not divide n.
%H A300715 Alois P. Heinz, <a href="/A300715/b300715.txt">Table of n, a(n) for n = 0..5000</a>
%H A300715 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A300715 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A300715 a(21) = 4 because we have [9, 4, 4, 4], [4, 9, 4, 4], [4, 4, 9, 4] and [4, 4, 4, 9].
%p A300715 a:= proc(m) option remember; local b; b:= proc(n) option
%p A300715       remember; `if`(n=0, 1, add((s->`if`(s>n or irem(m, s)
%p A300715        =0, 0, b(n-s)))(j^2), j=2..isqrt(n))) end; b(m)
%p A300715     end:
%p A300715 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 11 2018
%t A300715 Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0 && IntegerQ[k^(1/2)]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 75}]
%Y A300715 Cf. A000290, A006456, A294105, A294265, A294266, A300702, A300703, A300704, A300706.
%K A300715 nonn
%O A300715 0,14
%A A300715 _Ilya Gutkovskiy_, Mar 11 2018