A354734 a(0) = a(1) = 1; a(n) = 3 * Sum_{k=0..n-2} a(k) * a(n-k-2).

1, 1, 3, 6, 21, 54, 189, 558, 1944, 6210, 21681, 72576, 254988, 878850, 3112101, 10935000, 39030660, 139001346, 499808232, 1797731496, 6506661798, 23583173328, 85847830965, 313063862436, 1145325387114, 4197826175634, 15424343762184, 56774049331356, 209400739623054

Offset

0,3

Links

Table of n, a(n) for n=0..28.

Formula

G.f. A(x) satisfies: A(x) = 1 + x + 3 * (x * A(x))^2.

G.f.: (1 - sqrt(1 - 12 * x^2 * (1 + x))) / (6 * x^2).

a(n) ~ sqrt((2+3*r)*(1+r)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*cos(arccos(1/8)/3)/3 - 1/3. - Vaclav Kotesovec, Jun 04 2022

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = 3 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]
nmax = 28; CoefficientList[Series[(1 - Sqrt[1 - 12 x^2 (1 + x)])/(6 x^2), {x, 0, nmax}], x]

Crossrefs

Cf. A005159, A007477, A354733, A354735, A354736.

Sequence in context: A148611 A148612 A098511 * A112520 A148613 A148614

Adjacent sequences: A354731 A354732 A354733 * A354735 A354736 A354737

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 04 2022

Status

approved