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A029930 If 2n = Sum 2^e_i, a(n) = Product 2^e_i.

%I A029930 #37 Oct 31 2021 09:36:24
%S A029930 1,2,4,8,8,16,32,64,16,32,64,128,128,256,512,1024,32,64,128,256,256,
%T A029930 512,1024,2048,512,1024,2048,4096,4096,8192,16384,32768,64,128,256,
%U A029930 512,512,1024,2048,4096,1024,2048,4096,8192,8192,16384,32768,65536,2048
%N A029930 If 2n = Sum 2^e_i, a(n) = Product 2^e_i.
%H A029930 Seiichi Manyama, <a href="/A029930/b029930.txt">Table of n, a(n) for n = 0..8191</a>
%H A029930 Arvind Ayyer, A. Prasad and S. Spallone, <a href="http://arxiv.org/abs/1604.08837">Representations of symmetric groups with non-trivial determinant</a>, arXiv preprint arXiv:1604.08837 [math.RT], 2016. See Eq. (14).
%F A029930 From _Ralf Stephan_, Jun 19 2003: (Start)
%F A029930 G.f.: Prod_{k>=0} 1+2^(k+1)x^2^k.
%F A029930 a(0) = 1, a(2n) = 2^e1(n)*a(n), a(2n+1) = 2a(2n), where e1(n) = A000120(n).
%F A029930 a(n) = 2^A029931(n). (End)
%e A029930 14 = 8+4+2 so a(7) = 8*4*2 = 64.
%p A029930 HammingWeight := n -> add(i, i = convert(n, base, 2)):
%p A029930 a := proc(n) option remember; `if`(n = 0, 1,
%p A029930 ifelse(n::even, 2^HammingWeight(n/2)*a(n/2), 2*a(n-1))) end:
%p A029930 seq(a(n), n = 0..48); # _Peter Luschny_, Oct 30 2021
%t A029930 e1[n_] := Total[IntegerDigits[n, 2]]; a[0] = 1; a[n_] := a[n] = If[EvenQ[ n], 2^e1[n/2] a[n/2], 2 a[n-1]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Mar 07 2016 *)
%o A029930 (PARI) a(n) = {my(bd = Vecrev(binary(n))); prod(k=1, #bd, if (bd[k], 2^k, 1));} \\ _Michel Marcus_, Mar 07 2016
%Y A029930 Cf. A000120, A029931, A073642.
%Y A029930 A bisection of A059867.
%K A029930 nonn,easy,look
%O A029930 0,2
%A A029930 _N. J. A. Sloane_