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A382978 Expansion of Product_{k>=1} (1 + (2^k - 1) * x^k).

%I A382978 #23 Apr 14 2025 07:38:54
%S A382978 1,1,3,10,22,67,160,433,986,2774,6386,16214,39201,95868,229644,569707,
%T A382978 1324730,3186326,7664378,17955006,42497434,100710158,235492595,
%U A382978 549267552,1288847672,2990756088,6958113345,16148883002,37286262238,85880711282,198840926982,454980392570
%N A382978 Expansion of Product_{k>=1} (1 + (2^k - 1) * x^k).
%C A382978 This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -(2^n - 1).
%H A382978 Vincenzo Librandi, <a href="/A382978/b382978.txt">Table of n, a(n) for n = 0..600</a>
%H A382978 Seiichi Manyama, <a href="https://oeis.org/wiki/User:Seiichi_Manyama/Generalized_Euler_transform">Generalized Euler transform</a>.
%F A382978 a(n) = Sum_{k=0..n} 2^k * A382975(k,n-k).
%F A382978 a(n) ~ A048651 * 2^(n-1) * exp(Pi*sqrt(n/3)) / (3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 13 2025
%t A382978 n=31;CoefficientList[Normal@Series[Product[(1+(2^k-1) x^k),{k,1,n}],{x,0,n}],x] (* _Vincenzo Librandi_, Apr 11 2025 *)
%o A382978 (PARI) f(n) = -1;
%o A382978 g(n) = -(2^n-1);
%o A382978 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 A382978 (Magma) n := 31; R<x> := PowerSeriesRing(Rationals(), n+1); f := &*[ (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 A382978 Cf. A322199, A382976, A382977.
%Y A382978 Cf. A000225, A266964, A048651.
%K A382978 nonn
%O A382978 0,3
%A A382978 _Seiichi Manyama_, Apr 11 2025