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A053197 Number of level partitions of n.

%I A053197 #17 Jan 02 2016 18:47:05
%S A053197 1,1,2,2,4,3,6,5,10,8,13,12,21,18,27,27,42,38,54,54,77,76,101,104,143,
%T A053197 142,183,192,249,256,323,340,432,448,550,585,722,760,918,982,1190,
%U A053197 1260,1502,1610,1917,2048,2408,2590,3053,3264,3800,4097,4765,5120,5910,6378
%N A053197 Number of level partitions of n.
%C A053197 A partition is level if the powers of 2 dividing its parts are all equal.
%H A053197 Alois P. Heinz, <a href="/A053197/b053197.txt">Table of n, a(n) for n = 0..10000</a>
%F A053197 a(n) = Sum_{k=0..A007814(n)} A000009(n/2^k). a(2*n+1) = A000009(2*n+1) = A078408(n). - _Vladeta Jovovic_, Sep 29 2004
%p A053197 b:= proc(n, i, p) option remember; `if`(n=0, 1,
%p A053197      `if`(i<1, 0, add(b(n-i*j, i-p, p), j=0..n/i)))
%p A053197     end:
%p A053197 a:= n-> (m-> `if`(n=0, 1, add(b(n, (h-> h-1+irem(h, 2)
%p A053197     )(iquo(n, 2^j))*2^j, 2^(1+j)), j=0..m)))(ilog2(n)):
%p A053197 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jun 11 2015
%t A053197 a[n_] := Sum[ PartitionsQ[n/2^k], {k, 0, IntegerExponent[n, 2]}]; Table[ a[n], {n, 1, 55}] (* _Jean-François Alcover_, Dec 12 2011, after Vladeta Jovovic *)
%Y A053197 Cf. A049313, A053195.
%K A053197 nonn,nice
%O A053197 0,3
%A A053197 _Vladeta Jovovic_, Mar 02 2000
%E A053197 a(0)=1 prepended by _Alois P. Heinz_, Jun 11 2015