A346505 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^2) / (1 - x + 2 * x^2).
1, 2, 4, 12, 44, 172, 700, 2940, 12652, 55500, 247260, 1115740, 5088908, 23423020, 108659324, 507520316, 2384733868, 11264884876, 53464215580, 254822253852, 1219182031820, 5853309920748, 28190437248700, 136160853462524, 659401832797676, 3201141695492172, 15575294057678428
Offset: 0
Keywords
Programs
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Mathematica
nmax = 26; A[] = 0; Do[A[x] = (1 + x A[x]^2)/(1 - x + 2 x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[a[k] a[n - k - 1], {k, 2, n - 1}]; Table[a[n], {n, 0, 26}] CoefficientList[Series[(1 - x*(1 - 2*x)) * (1 - Sqrt[1 - 4*x/(-1 + x - 2*x^2)^2]) / (2*x), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 26 2023 *)
Formula
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=2..n-1} a(k) * a(n-k-1).
From Vaclav Kotesovec, Sep 26 2023: (Start)
G.f.: (1 - x*(1 - 2*x))*(1 - sqrt(1 - 4*x/(-1 + x - 2*x^2)^2))/(2*x).
a(n) ~ sqrt((69 + 57*sqrt(114) + 23*3^(5/6)*sqrt(38)*(9 + 2*sqrt(114))^(1/3) - 36*3^(1/6)*sqrt(38)*(9 + 2*sqrt(114))^(2/3) + 291*(27 + 6*sqrt(114))^(1/3) - 54*(27 + 6*sqrt(114))^(2/3))/(-72 - 16*sqrt(114) + 3^(11/6)*sqrt(38)*(9 + 2*sqrt(114))^(1/3) + 3^(1/6)*sqrt(38)*(9 + 2*sqrt(114))^(2/3) + 26*(27 + 6*sqrt(114))^(1/3) - 6*(27 + 6*sqrt(114))^(2/3))) * 2^(n - 1/2) * 3^(1/6 + 4*n/3) * ((9 + 2*sqrt(114))^((1/3)*(n-1)) / (sqrt(Pi) * n^(3/2) * (-15 + (27 + 6*sqrt(114))^(2/3))^n)). (End)