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A029930 If 2n = Sum 2^e_i, a(n) = Product 2^e_i. 7

%I #37 Oct 31 2021 09:36:24

%S 1,2,4,8,8,16,32,64,16,32,64,128,128,256,512,1024,32,64,128,256,256,

%T 512,1024,2048,512,1024,2048,4096,4096,8192,16384,32768,64,128,256,

%U 512,512,1024,2048,4096,1024,2048,4096,8192,8192,16384,32768,65536,2048

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

%H Seiichi Manyama, <a href="/A029930/b029930.txt">Table of n, a(n) for n = 0..8191</a>

%H 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 From _Ralf Stephan_, Jun 19 2003: (Start)

%F G.f.: Prod_{k>=0} 1+2^(k+1)x^2^k.

%F a(0) = 1, a(2n) = 2^e1(n)*a(n), a(2n+1) = 2a(2n), where e1(n) = A000120(n).

%F a(n) = 2^A029931(n). (End)

%e 14 = 8+4+2 so a(7) = 8*4*2 = 64.

%p HammingWeight := n -> add(i, i = convert(n, base, 2)):

%p a := proc(n) option remember; `if`(n = 0, 1,

%p ifelse(n::even, 2^HammingWeight(n/2)*a(n/2), 2*a(n-1))) end:

%p seq(a(n), n = 0..48); # _Peter Luschny_, Oct 30 2021

%t 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 (PARI) a(n) = {my(bd = Vecrev(binary(n))); prod(k=1, #bd, if (bd[k], 2^k, 1));} \\ _Michel Marcus_, Mar 07 2016

%Y Cf. A000120, A029931, A073642.

%Y A bisection of A059867.

%K nonn,easy,look

%O 0,2

%A _N. J. A. Sloane_

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Last modified March 28 07:48 EDT 2024. Contains 371235 sequences. (Running on oeis4.)