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A382977 Expansion of Product_{k>=1} 1/(1 - (2^k - 1) * x^k).

%I A382977 #21 Apr 13 2025 16:21:32
%S A382977 1,1,4,11,35,87,271,659,1908,4832,13132,32688,89109,218385,571489,
%T A382977 1427388,3652877,8980805,22858201,55822728,140065621,342001192,
%U A382977 845707856,2052802367,5057431745,12197383588,29738238996,71604414162,173406091548,415167136507,1000881376700
%N A382977 Expansion of Product_{k>=1} 1/(1 - (2^k - 1) * x^k).
%C A382977 This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^n - 1.
%H A382977 Vincenzo Librandi, <a href="/A382977/b382977.txt">Table of n, a(n) for n = 0..600</a>
%H A382977 Seiichi Manyama, <a href="https://oeis.org/wiki/User:Seiichi_Manyama/Generalized_Euler_transform">Generalized Euler transform</a>.
%F A382977 a(n) = Sum_{k=0..n} 2^k * A382974(k,n-k).
%F A382977 log(a(n)) ~ n*log(2) + Pi*sqrt(2*n/3). - _Vaclav Kotesovec_, Apr 13 2025
%t A382977 n=30; CoefficientList[Normal@Series[Product[1/(1-(2^k-1) x^k),{k,1,n}],{x,0,n}],x] (* _Vincenzo Librandi_, Apr 11 2025 *)
%o A382977 (PARI) f(n) = 1;
%o A382977 g(n) = 2^n-1;
%o A382977 a_vector(n) = my(b=vector(n, k, sumdiv(k, d, d*f(d)*g(d)^(k/d))), v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, b[j]*v[i-j+1])/i); v;
%o A382977 (Magma)  n := 30; R<x> := PowerSeriesRing(Rationals(), n+1); f := &*[ 1 / (1 - (2^k - 1)*x^k) : k in [1..n] ]; coeffs := [Coefficient(f, i) : i in [0..n]]; coeffs; // _Vincenzo Librandi_, Apr 11 2025
%Y A382977 Cf. A322199, A382976, A382978.
%Y A382977 Cf. A000225, A266964.
%K A382977 nonn
%O A382977 0,3
%A A382977 _Seiichi Manyama_, Apr 11 2025