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A349253
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G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 2 * x * A(x)^2)).
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6
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1, 3, 19, 169, 1753, 19795, 236035, 2923857, 37256881, 485202307, 6429346899, 86405569657, 1174917167881, 16134949855251, 223460304878467, 3117521211476641, 43771643214792033, 618045740600046211, 8770377489446217235, 125013010654218317385, 1789104455068153153849
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 1 + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * 2^k * binomial(3*k,k) / (2*k+1).
a(n) = hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], -(3/2)^3). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(315 + 31*sqrt(105)) * (31 + 3*sqrt(105))^n / (9 * sqrt(Pi) * 2^(2*n + 5/2) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021
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MATHEMATICA
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nmax = 20; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 20}]
a[n_] := HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, -(3/2)^3];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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