A339447 - OEIS

%I #15 Dec 05 2020 14:43:21

%S 0,1,2,2,4,4,6,7,11,12,17,20,27,32,42,50,66,78,99,119,150,179,223,267,

%T 329,393,480,572,695,826,995,1181,1417,1674,1997,2355,2796,3288,3887,

%U 4558,5370,6281,7371,8603,10067,11719,13674,15886,18486,21432,24879,28787,33344

%N G.f.: Sum_{k>=0} x^(2^k) / Product_{j=1..2^k} (1 - x^j).

%C Number of partitions of n such that the number of parts is a power of 2.

%C Also number of partitions of n such that the largest part is a power of 2.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%e a(6) = 6 because we have [6], [5, 1], [4, 2], [3, 3], [3, 1, 1, 1] and [2, 2, 1, 1] (see the first comment) or [4, 2], [4, 1, 1], [2, 2, 2], [2, 2, 1, 1], [2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1] (see the second comment).

%t nmax = 52; CoefficientList[Series[Sum[x^(2^k)/Product[1 - x^j, {j, 1, 2^k}], {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

%Y Cf. A000079, A018819.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 05 2020