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A370260
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a(n) = sqrt(A370259(2*n+1)).
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5
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1, 3, 31, 617, 18529, 748859, 38149567, 2348482961, 169641143873, 14071599763379, 1318414335714015, 137720427724123513, 15871136311527376801, 2000355821099358166891, 273735526097742996298111, 40419227378551955037029921, 6405616571975691389276400257
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OFFSET
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0,2
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COMMENTS
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The sequence is conjectured to be integral [added 03 Mar 2024: now confirmed - see the Formula section].
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LINKS
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FORMULA
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a(n) = sqrt( (T(2*n+1, 2*n+2) - 1)/(2*n+1)^3 ), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
a(n) = sqrt( Sum_{k = 1..2*n+1} (2^k)*(2*n + 1)^(k-2)*binomial(2*n + k + 1, 2*k)/(2*n + k + 1) ).
a(n) = Sum_{k = 0..n} binomial(n+k, n-k)/(2*k + 1) * (4*n + 2)^k (shows the sequence to be integral) = R(n,2), where R(n, x) is the n-th row polynomial of A370262. - Peter Bala, Apr 03 2024
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MAPLE
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A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n+1)), n = 0..20);
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MATHEMATICA
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Table[Sqrt[(ChebyshevT[k, k + 1] - 1)/k^3], {k, 1, 40, 2}] (* Paolo Xausa, Jul 24 2024 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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