cp's OEIS Frontend

A327683 Expansion of Product_{k>0} (1+sqrt(1+4*x^k))/2.

%I A327683 #39 May 06 2021 04:34:53
%S A327683 1,1,0,4,-5,17,-40,144,-459,1517,-5111,17747,-62074,219292,-782602,
%T A327683 2816664,-10205754,37203230,-136360106,502219652,-1857659296,
%U A327683 6897983144,-25704335380,96090440940,-360265425619,1354343161419,-5103948546609,19278502980063,-72972099256954
%N A327683 Expansion of Product_{k>0} (1+sqrt(1+4*x^k))/2.
%H A327683 Seiichi Manyama, <a href="/A327683/b327683.txt">Table of n, a(n) for n = 0..1000</a>
%F A327683 a(n) ~ -(-1)^n * c * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + sqrt(1 + 4*(-1/4)^k))/2 = 0.52271977595412566689522667777276363119313248923... - _Vaclav Kotesovec_, May 06 2021
%p A327683 N:= 40:
%p A327683 P:= mul((1+sqrt(1+4*x^k))/2,k=1..N):
%p A327683 S:= series(P,x,N+1):
%p A327683 seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Sep 22 2019
%t A327683 nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1+4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 22 2019 *)
%o A327683 (PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1+4*x^k))/2))
%o A327683 (PARI) N=66; x='x+O('x^N); Vec(prod(i=1, N, 1-sum(j=1, N\i, (-1)^j*binomial(2*j-2, j-1)*x^(i*j)/j)))
%Y A327683 Convolution inverse of A327682.
%Y A327683 Cf. A000009, A268498, A322204, A327684.
%K A327683 sign
%O A327683 0,4
%A A327683 _Seiichi Manyama_, Sep 22 2019