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A305652 Expansion of Product_{k>=1} (1 + x^k)^(2^(k-1)-1).

%I A305652 #10 Jun 09 2018 00:24:14
%S A305652 1,0,1,3,7,18,41,99,227,538,1236,2872,6597,15166,34669,79150,180011,
%T A305652 408616,925015,2089607,4709937,10595275,23788174,53312366,119271967,
%U A305652 266399612,594077742,1322815256,2941225084,6530659320,14481362803,32070677496,70937233268,156721128440
%N A305652 Expansion of Product_{k>=1} (1 + x^k)^(2^(k-1)-1).
%C A305652 Weigh transform of A000225, shifted right one place.
%C A305652 Convolution of the sequences A081362 and A098407.
%H A305652 Alois P. Heinz, <a href="/A305652/b305652.txt">Table of n, a(n) for n = 0..1000</a>
%H A305652 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305652 G.f.: Product_{k>=1} (1 + x^k)^A000225(k-1).
%F A305652 G.f.: Product_{k>=1} (1 + x^k)^(A011782(k)-1).
%F A305652 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^(2*k)/(k*(1 - x^k)*(1 - 2*x^k))).
%F A305652 a(n) ~ 2^n * exp(sqrt(2*n) - 5/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-1)*(2^k-2)) = -0.07640757130267274170429705262846... - _Vaclav Kotesovec_, Jun 08 2018
%p A305652 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305652       add(binomial(2^(i-1)-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p A305652     end:
%p A305652 a:= n-> b(n$2):
%p A305652 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jun 07 2018
%t A305652 nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(2^(k-1)-1), {k, 1, nmax}], {x, 0, nmax}], x]
%t A305652 nmax = 33; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^(2 k)/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A305652 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^(d - 1) - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]
%Y A305652 Cf. A000225, A001906, A011782, A081362, A098407, A102866, A110045.
%K A305652 nonn
%O A305652 0,4
%A A305652 _Ilya Gutkovskiy_, Jun 07 2018