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A304393 Expansion of Product_{k>0} (1 + Sum_{m>=0} x^(k*2^m)).

%I A304393 #30 Feb 14 2023 06:30:50
%S A304393 1,1,2,2,5,5,8,10,17,19,27,33,48,56,76,92,126,146,192,228,298,352,444,
%T A304393 528,667,783,969,1145,1414,1658,2017,2365,2878,3352,4027,4703,5634,
%U A304393 6548,7773,9033,10705,12381,14573,16857,19790,22800,26631,30655,35723,41005
%N A304393 Expansion of Product_{k>0} (1 + Sum_{m>=0} x^(k*2^m)).
%C A304393 Also the number of partitions of n in which each part occurs a power of 2 (cf. A000079) of times.
%H A304393 Vaclav Kotesovec, <a href="/A304393/b304393.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)
%e A304393 n | Partitions of n in which each part occurs a power of 2 (cf. A000079) of times
%e A304393 --+------------------------------------------------------------------------------
%e A304393 1 | 1;
%e A304393 2 | 2 = 1+1;
%e A304393 3 | 3 = 2+1;
%e A304393 4 | 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1;
%e A304393 5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1;
%e A304393 6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 2+1+1+1+1;
%e A304393 7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1;
%p A304393 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A304393       b(n, i-1)+add(b(n-i*2^j, i-1), j=0..ilog2(n/i))))
%p A304393     end:
%p A304393 a:= n-> b(n$2):
%p A304393 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 13 2018
%t A304393 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A304393    b[n, i-1] + Sum[b[n-i*2^j, i-1], {j, 0, Floor@Log2[n/i]}]]];
%t A304393 a[n_] := b[n, n];
%t A304393 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 14 2023, after _Alois P. Heinz_ *)
%Y A304393 Cf. A000079, A055922, A300446, A304332.
%K A304393 nonn
%O A304393 0,3
%A A304393 _Seiichi Manyama_, May 12 2018