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A349289
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G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^3)).
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7
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1, 2, 9, 61, 493, 4371, 41065, 401563, 4044097, 41658044, 436862457, 4648331765, 50057856881, 544557984498, 5975422922413, 66059269445451, 735064865871889, 8226310738656892, 92531697191189777, 1045551973586825023, 11862334695799444993
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OFFSET
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0,2
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..500
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FORMULA
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a(n) = Sum_{k=0..n} binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1).
a(n) = F([1/4, 1/2, 3/4, (1+n)/2, (2+n)/2, -n], [1/3, 2/3, 2/3, 1, 4/3], -2^10/3^6) where F is the generalized hypergeometric function. - Stefano Spezia, Nov 13 2021
a(n) ~ sqrt(1 + 2*r) / (4 * 2^(1/6) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/3)), where r = 0.0816785448577670972635343365300887975661075663022821172271... is the root of the equation 4^4 * r = 3^3 * (1-r)^3. - Vaclav Kotesovec, Nov 14 2021
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MATHEMATICA
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nmax = 20; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]
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CROSSREFS
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Cf. A002293, A007317, A199475, A346646, A349290, A349291, A349292, A349293.
Sequence in context: A207649 A289713 A341956 * A053983 A192939 A107883
Adjacent sequences: A349286 A349287 A349288 * A349290 A349291 A349292
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, Nov 13 2021
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STATUS
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approved
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