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A336165 G.f. A(x) satisfies: A(x) = 1 + x * ((1 - x) * A(x))^2. 0

%I

%S 1,1,0,-2,-4,-3,6,26,46,22,-128,-455,-748,-149,2948,9400,14254,-1624,

%T -72876,-212988,-294316,143030,1889284,5104273,6328244,-6017051,

%U -50569884,-126812057,-138104146,216071703,1383709740,3226295732,2992392698,-7280984690

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

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

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

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

%F 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)).

%t nmax = 33; CoefficientList[Series[(1 - Sqrt[1 - 4 x (1 - x)^2])/(2 x (1 - x)^2), {x, 0, nmax}], x]

%t Table[Sum[(-1)^(n - k) Binomial[2 k, n - k] CatalanNumber[k], {k, 0, n}], {n, 0, 33}]

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

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Jul 10 2020

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Last modified June 22 14:27 EDT 2021. Contains 345380 sequences. (Running on oeis4.)