A336165 G.f. A(x) satisfies: A(x) = 1 + x * ((1 - x) * A(x))^2.

1, 1, 0, -2, -4, -3, 6, 26, 46, 22, -128, -455, -748, -149, 2948, 9400, 14254, -1624, -72876, -212988, -294316, 143030, 1889284, 5104273, 6328244, -6017051, -50569884, -126812057, -138104146, 216071703, 1383709740, 3226295732, 2992392698, -7280984690

Offset

0,4

Links

Table of n, a(n) for n=0..33.

Formula

G.f.: (1 - sqrt(1 - 4*x*(1 - x)^2)) / (2*x*(1 - x)^2).

G.f.: 1 / (1 - x*(1 - x)^2 / (1 - x*(1 - x)^2 / (1 - x*(1 - x)^2 / (1 - ...)))), a continued fraction.

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,n-k) * A000108(k).

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

D-finite with recurrence (n+1)*a(n) -5*n*a(n-1) +2*(6*n-7)*a(n-2) +2*(-6*n+11)*a(n-3) +2*(2*n-5)*a(n-4)=0. - R. J. Mathar, Mar 06 2022

Mathematica

nmax = 33; CoefficientList[Series[(1 - Sqrt[1 - 4 x (1 - x)^2])/(2 x (1 - x)^2), {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[2 k, n - k] CatalanNumber[k], {k, 0, n}], {n, 0, 33}]

Crossrefs

Cf. A000108, A006319, A073155, A115399, A256169.

Sequence in context: A122525 A054589 A051851 * A011171 A181049 A007203

Adjacent sequences: A336162 A336163 A336164 * A336166 A336167 A336168

Keyword

sign,easy

Author

Ilya Gutkovskiy, Jul 10 2020

Status

approved