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

1, 1, 0, 4, -5, 17, -40, 144, -459, 1517, -5111, 17747, -62074, 219292, -782602, 2816664, -10205754, 37203230, -136360106, 502219652, -1857659296, 6897983144, -25704335380, 96090440940, -360265425619, 1354343161419, -5103948546609, 19278502980063, -72972099256954

Offset: 0

Author

Seiichi Manyama, Sep 22 2019

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Convolution inverse of A327682.

Cf. A000009, A268498, A322204, A327684.

Programs

Maple

N:= 40:
P:= mul((1+sqrt(1+4*x^k))/2,k=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Sep 22 2019

Mathematica

nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1+4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 22 2019 *)

PARI

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1+4*x^k))/2))

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)))

Formula

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