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A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).

%I A319919 #13 Dec 14 2024 09:14:15
%S A319919 1,1,3,10,25,70,182,476,1220,3122,7883,19794,49340,122237,301114,
%T A319919 737923,1799597,4369204,10563800,25441377,61048713,145988775,
%U A319919 347981713,826921992,1959363778,4629903905,10911757432,25652950459,60165831361,140792215037,328750398275,766041930160,1781452975346
%N A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).
%C A319919 Convolution of A081362 and A102866.
%C A319919 Weigh transform of A000225.
%H A319919 Vaclav Kotesovec, <a href="/A319919/b319919.txt">Table of n, a(n) for n = 0..3000</a>
%H A319919 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A319919 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)*(1 - 2*x^k))).
%F A319919 a(n) ~ c * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A079555 * sqrt(Pi) * n^(3/4)), where c = exp(Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1))) = 0.6602994483152065685... - _Vaclav Kotesovec_, Sep 15 2021
%p A319919 a:=series(mul((1+x^k)^(2^k-1),k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # _Paolo P. Lava_, Apr 02 2019
%t A319919 nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
%t A319919 nmax = 32; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A319919 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
%Y A319919 Cf. A000225, A007070, A081362, A098407, A102866, A319918.
%K A319919 nonn
%O A319919 0,3
%A A319919 _Ilya Gutkovskiy_, Oct 01 2018