A346506 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^2) / (1 - x + x^2).
1, 2, 5, 17, 66, 274, 1190, 5341, 24577, 115326, 549747, 2654739, 12959468, 63848307, 317064921, 1585380283, 7975134892, 40332823042, 204947059412, 1045859173864, 5357606584326, 27540884494209, 142023060613755, 734506610474205, 3808771672620618, 19798640525731461, 103149287155802941
Offset: 0
Keywords
Programs
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Mathematica
nmax = 26; A[] = 0; Do[A[x] = (1 + x A[x]^2)/(1 - x + x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] a[0] = 1; a[1] = 2; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 2, n - 1}]; Table[a[n], {n, 0, 26}]
Formula
a(0) = 1, a(1) = 2; a(n) = 2 * a(n-1) + a(n-2) + Sum_{k=2..n-1} a(k) * a(n-k-1).
From Nikolaos Pantelidis, Jan 08 2023 (Start)
G.f.: 1/G(0), where G(k) = 1-(2*x-x^2)/(1-x/G(k+1)) (continued fraction).
G.f.: (1-x+x^2-sqrt(x^4-2*x^3+3*x^2-6*x+1))/(2*x).
(End)