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