A331923 - OEIS

%I #14 Oct 25 2020 08:51:24

%S 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 7,10,32,8,11,44,150,30,34,40,18,26,20,68,78,126,56,169,80,30,40,116,

%U 294,144,162,226,182,128,66,338,348,752,199,1048

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

%H Robert Israel, <a href="/A331923/b331923.txt">Table of n, a(n) for n = 0..2600</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%e a(17) = 4 because we have [16, 1], [9, 8], [8, 9] and [1, 16].

%p N:= 200: # for a(0)..a(N)

%p PP:= {1,seq(seq(b^i,i=2..floor(log[b](N))),b=2..floor(sqrt(N)))}:

%p G:= mul(1+t*x^p, p=PP):

%p F:= proc(n) local R, k, v;

%p   R:= normal(coeff(G, x, n));

%p   add(k!*coeff(R, t, k), k=1..degree(R, t))

%p end proc:

%p F(0):= 1:

%p map(F, [$0..N]); # _Robert Israel_, Feb 03 2020

%t M = 200;

%t PP = Join[{1}, Table[Table[b^i, {i, 2, Floor[Log[b, M]]}], {b, 2, Floor[ Sqrt[M]]}] // Flatten // Union];

%t G = Product[1 + t x^p, {p, PP}];

%t 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 a[0] = 1;

%t a /@ Range[0, M] (* _Jean-François Alcover_, Oct 25 2020, after _Robert Israel_ *)

%Y Cf. A001597, A078635, A112345, A282500, A284171, A331844, A331845.

%K nonn,look

%O 0,6

%A _Ilya Gutkovskiy_, Feb 01 2020