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A331923 Number of compositions (ordered partitions) of n into distinct perfect powers.

%I A331923 #14 Oct 25 2020 08:51:24
%S A331923 1,1,0,0,1,2,0,0,1,3,2,0,2,8,6,0,1,4,6,0,2,12,24,0,2,9,8,1,8,32,30,2,
%T A331923 7,10,32,8,11,44,150,30,34,40,18,26,20,68,78,126,56,169,80,30,40,116,
%U A331923 294,144,162,226,182,128,66,338,348,752,199,1048
%N A331923 Number of compositions (ordered partitions) of n into distinct perfect powers.
%H A331923 Robert Israel, <a href="/A331923/b331923.txt">Table of n, a(n) for n = 0..2600</a>
%H A331923 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%e A331923 a(17) = 4 because we have [16, 1], [9, 8], [8, 9] and [1, 16].
%p A331923 N:= 200: # for a(0)..a(N)
%p A331923 PP:= {1,seq(seq(b^i,i=2..floor(log[b](N))),b=2..floor(sqrt(N)))}:
%p A331923 G:= mul(1+t*x^p, p=PP):
%p A331923 F:= proc(n) local R, k, v;
%p A331923   R:= normal(coeff(G, x, n));
%p A331923   add(k!*coeff(R, t, k), k=1..degree(R, t))
%p A331923 end proc:
%p A331923 F(0):= 1:
%p A331923 map(F, [$0..N]); # _Robert Israel_, Feb 03 2020
%t A331923 M = 200;
%t A331923 PP = Join[{1}, Table[Table[b^i, {i, 2, Floor[Log[b, M]]}], {b, 2, Floor[ Sqrt[M]]}] // Flatten // Union];
%t A331923 G = Product[1 + t x^p, {p, PP}];
%t A331923 a[n_] := Module[{R, k, v}, R = SeriesCoefficient[G, {x, 0, n}]; Sum[k! SeriesCoefficient[R, {t, 0, k}], {k, 1, Exponent[R, t]}]];
%t A331923 a[0] = 1;
%t A331923 a /@ Range[0, M] (* _Jean-François Alcover_, Oct 25 2020, after _Robert Israel_ *)
%Y A331923 Cf. A001597, A078635, A112345, A282500, A284171, A331844, A331845.
%K A331923 nonn,look
%O A331923 0,6
%A A331923 _Ilya Gutkovskiy_, Feb 01 2020