A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

1, 1, 4, 8, 36, 96, 416, 1312, 5504, 19200, 79168, 293888, 1203712, 4648448, 19027968, 75411456, 309487616, 1248411648, 5144133632, 21011775488, 86971449344, 358540509184, 1490753372160, 6189315784704, 25843660619776, 107902536122368, 452308820819968, 1897178275512320

Offset: 0

Author

Ilya Gutkovskiy, Jun 04 2022

Keywords

• nonn

Crossrefs

Cf. A007477, A052704, A151403, A354733, A354734, A354736.

Programs

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = 4 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]
nmax = 27; CoefficientList[Series[(1 - Sqrt[1 - 16 x^2 (1 + x)])/(8 x^2), {x, 0, nmax}], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x + 4 * (x * A(x))^2.

G.f.: (1 - sqrt(1 - 16 * x^2 * (1 + x))) / (8 * x^2).

a(n) ~ sqrt((2+3*r)*(1+r)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*cos(arccos(-5/32)/3)/3 - 1/3. - Vaclav Kotesovec, Jun 04 2022