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A289842 Sum of products of terms in all partitions of 2*n into powers of 2.

%I A289842 #51 Nov 23 2020 06:56:56
%S A289842 1,3,11,27,83,195,515,1155,2899,6387,15219,32883,76275,163059,368883,
%T A289842 780531,1738259,3653715,8022355,16759635,36428371,75765843,163217491,
%U A289842 338120787,723384915,1493913171,3176799827,6542573139,13844246099,28447592019,59934789203
%N A289842 Sum of products of terms in all partitions of 2*n into powers of 2.
%H A289842 Seiichi Manyama, <a href="/A289842/b289842.txt">Table of n, a(n) for n = 0..3309</a>
%F A289842 a(n) = [x^(2*n)] Product_{k>=0} 1/(1 - 2^k*x^(2^k)). - _Ilya Gutkovskiy_, Sep 10 2018
%F A289842 a(n) ~ c * n * 2^n, where c = 2.1343755406794500897789546611306737041750472866941557748356... - _Vaclav Kotesovec_, Jun 18 2019
%e A289842 n | partitions of 2*n into powers of 2                 | a(n)
%e A289842 --------------------------------------------------------------------------
%e A289842 1 | 2  , 1+1                                           | 2+1         =  3.
%e A289842 2 | 4  , 2+2  , 2+1+1, 1+1+1+1                         | 4+4+2+1     = 11.
%e A289842 3 | 4+2, 4+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1 | 8+4+8+4+2+1 = 27.
%p A289842 b:= proc(n, i, p) option remember; `if`(n=0, p,
%p A289842      `if`(i<1, 0, add(b(n-j*i, i/2, p*i^j), j=0..n/i)))
%p A289842     end:
%p A289842 a:= n-> (t-> b(t, 2^ilog2(t), 1))(2*n):
%p A289842 seq(a(n), n=0..33);  # _Alois P. Heinz_, Oct 27 2017
%t A289842 b[n_, i_, p_] := b[n, i, p] = If[n == 0, p, If[i < 1, 0, Sum[b[n - j i, i/2, p i^j], {j, 0, n/i}]]];
%t A289842 a[n_] := b[2n, 2^Floor@Log[2, 2n], 1];
%t A289842 a /@ Range[0, 33] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%Y A289842 Cf. A000123, A018819.
%K A289842 nonn
%O A289842 0,2
%A A289842 _Seiichi Manyama_, Oct 27 2017