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A349254
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G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - 3 * x * A(x)^2)).
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4
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1, 4, 37, 478, 7159, 116497, 2000386, 35671756, 654218641, 12261271942, 233798163646, 4521194100541, 88458184054882, 1747850650032532, 34828329987024058, 699083528482636228, 14121906499195594537, 286877562430915732546, 5856866441794110926809
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..18.
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FORMULA
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a(n) = 1 + 3 * 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) * 3^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)^4). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(873 + 89*sqrt(97)) * (89 + 9*sqrt(97))^n / (3^(5/2) * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021
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MATHEMATICA
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nmax = 18; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - 3 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + 3 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, 18}]
Table[Sum[Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 18}]
a[n_] := HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, -81/16];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Nov 12 2021 *)
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CROSSREFS
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Cf. A001764, A103211, A199475, A217363, A337167, A349253, A349255, A349256.
Sequence in context: A221630 A235135 A316877 * A277638 A352237 A349714
Adjacent sequences: A349251 A349252 A349253 * A349255 A349256 A349257
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, Nov 12 2021
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STATUS
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approved
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