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A382976 Expansion of Product_{k>=1} (1 + (2^k + 1) * x^k).

%I A382976 #22 Apr 11 2025 11:16:15
%S A382976 1,3,5,24,44,129,384,897,2220,5706,15268,35178,89829,212982,526222,
%T A382976 1294263,3087570,7300896,17726100,41705904,98782950,236059794,
%U A382976 551697495,1293417672,3033232130,7081297146,16430673765,38347412562,88762751808,204970377366,473719894598
%N A382976 Expansion of Product_{k>=1} (1 + (2^k + 1) * x^k).
%C A382976 This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -(2^n + 1).
%H A382976 Vincenzo Librandi, <a href="/A382976/b382976.txt">Table of n, a(n) for n = 0..600</a>
%H A382976 Seiichi Manyama, <a href="https://oeis.org/wiki/User:Seiichi_Manyama/Generalized_Euler_transform">Generalized Euler transform</a>.
%F A382976 a(n) = Sum_{k=0..n} 2^k * A284593(k,n-k).
%F A382976 a(n) ~ A079555 * 2^(n-1) * exp(Pi*sqrt(n/3)) / (3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 11 2025
%t A382976 n=30; CoefficientList[Normal@Series[Product[1+(2^k+1) x^k,{k,1,n}],{x,0,n}],x] (* _Vincenzo Librandi_, Apr 11 2025 *)
%o A382976 (PARI) f(n) = -1;
%o A382976 g(n) = -(2^n+1);
%o A382976 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;
%Y A382976 Cf. A322199, A382977, A382978.
%Y A382976 Cf. A000051, A266964, A284593.
%Y A382976 Cf. A079555.
%K A382976 nonn
%O A382976 0,2
%A A382976 _Seiichi Manyama_, Apr 11 2025