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A357312 Number of compositions (ordered partitions) of n into divisors of n that are smaller than sqrt(n).

%I A357312 #10 Sep 24 2022 13:29:04
%S A357312 1,0,1,1,1,1,13,1,34,1,89,1,927,1,610,189,1597,1,35890,1,46754,1873,
%T A357312 28657,1,3919944,1,196418,18560,4205249,1,110187694,1,39882198,183916,
%U A357312 9227465,9496,10312882481,1,63245986,1822473,11969319436,1,141930520462,1,34020543362,339200673
%N A357312 Number of compositions (ordered partitions) of n into divisors of n that are smaller than sqrt(n).
%H A357312 Alois P. Heinz, <a href="/A357312/b357312.txt">Table of n, a(n) for n = 0..2000</a>
%F A357312 a(n) = [x^n] 1 / (1 - Sum_{d|n, d < sqrt(n)} x^d).
%p A357312 a:= proc(n) option remember; uses numtheory; local b, l;
%p A357312       l, b:= select(x-> is(x<sqrt(n)), divisors(n)),
%p A357312       proc(m) option remember; `if`(m=0, 1,
%p A357312          add(`if`(j>m, 0, b(m-j)), j=l))
%p A357312       end; b(n)
%p A357312     end:
%p A357312 seq(a(n), n=0..45);  # _Alois P. Heinz_, Sep 23 2022
%t A357312 a[n_] := SeriesCoefficient[1/(1 - Sum[Boole[d < Sqrt[n]] x^d, {d, Divisors[n]}]), {x, 0, n}]; Table[a[n], {n, 0, 45}]
%Y A357312 Cf. A100346, A294137, A294138, A327766, A357311.
%K A357312 nonn,look
%O A357312 0,7
%A A357312 _Ilya Gutkovskiy_, Sep 23 2022