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 A246138 a(n) = (Sum_{k=0..n-1} A246065(k)) / n^2. 8
 -1, 0, 1, 3, 9, 32, 135, 648, 3409, 19176, 113535, 700125, 4463415, 29256120, 196334697, 1344542787, 9371335905, 66335058128, 476022873279, 3457886816997, 25394948961831, 188353304179920, 1409578821465129, 10635308054118792, 80845157085234975 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Part (ii) of the conjecture in A246065 implies that all the terms in the current sequence are integers. Conjecture: The sequence a(n+1)/a(n) (n = 4,5,...) is strictly increasing to the limit 9, and the sequence a(n+1)^(1/(n+1))/a(n)^(1/n) (n = 3,4,...) is strictly decreasing to the limit 1. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..170 FORMULA Recurrence: n^2*a(n) = 2*(n-2)*(5*n-8)*a(n-1) - 9*(n-2)^2*a(n-2). - Vaclav Kotesovec, Aug 27 2014 a(n) ~ 3^(2*n+5/2) / (128*Pi*n^4). - Vaclav Kotesovec, Aug 27 2014 a(n) = ((3*n+2)*(3*n-2)*A005802(n-1) - (n+2)^2*A005802(n))/4. - Mark van Hoeij, Nov 06 2023 EXAMPLE a(5) = 9 since sum_{k=0}^{5-1}A246065(k) = -1 + 1 + 9 + 39 + 177 = 225 = 5^2*9. MAPLE ogf := (1-((9*x-1)/(x-1))^(3/4)*hypergeom([-1/4, 3/4], [1], -64*x/(9*x-1)^3/(x-1)))/6; series(ogf, x=0, 25); # Mark van Hoeij, Nov 12 2023 MATHEMATICA s[n_]:=Sum[Binomial[n, k]^2*Binomial[2k, k]/(2k-1), {k, 0, n}] a[n_]:=Sum[s[k], {k, 0, n-1}]/n^2 Table[a[n], {n, 1, 25}] CROSSREFS Cf. A005802, A246065. Sequence in context: A194530 A324238 A005964 * A129416 A210689 A334642 Adjacent sequences: A246135 A246136 A246137 * A246139 A246140 A246141 KEYWORD sign AUTHOR Zhi-Wei Sun, Aug 25 2014 STATUS approved

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Last modified August 11 04:26 EDT 2024. Contains 375059 sequences. (Running on oeis4.)