A329806 Expansion of Product_{k>=1} 1 / (1 - 6*x^k + x^(2*k))^(1/2).

1, 3, 16, 75, 385, 1971, 10473, 56139, 305394, 1674198, 9245506, 51325206, 286210243, 1601822505, 8992732043, 50619114252, 285583525237, 1614439389711, 9142794839933, 51858472602546, 294559269778199, 1675240507900632, 9538522900076376, 54367531265208579, 310179797595736539

Offset: 0

Author

Ilya Gutkovskiy, Nov 21 2019

Keywords

• nonn

Crossrefs

Cf. A001850, A067855.

Programs

Mathematica

nmax = 24; CoefficientList[Series[Product[1/(1 - 6 x^k + x^(2 k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[d (6 - x^d)^(k/d), {d, Divisors[k]}] x^k/(2 k), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d * (6 - x^d)^(k/d) ) * x^k / (2*k)).

G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A001850 (central Delannoy numbers).

a(n) ~ sqrt(2) * (1 + sqrt(2))^(2*n - 1/2) / (c * sqrt(Pi*n)), where c = QPochhammer[1/(1 + sqrt(2))^2] = 0.799142925985081767883272500537236047... - Vaclav Kotesovec, Nov 21 2019