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 A254747 a(n) = (1 + Sum_{j=0..n} (C(n,j)*C(3*j-1,j))) / 2. 1
 1, 2, 8, 47, 312, 2162, 15311, 109965, 797824, 5833298, 42910998, 317224800, 2354712927, 17538747124, 131017428431, 981194304302, 7364370502896, 55380344444150, 417176211054422, 3147365470080480, 23777750075552262 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA G.f.: x*G'(x)/G(x), where G(x) = x*(2/sqrt(3*x*(1-x)))*sin((1/3)*asin(3/2*sqrt(3*x/(1-x)))). a(n) = (hypergeom([1/3,2/3,-n],[1/2,1],-27/4)+2)/3. - Peter Luschny, Feb 07 2015 From Vaclav Kotesovec, Feb 07 2015: (Start) Recurrence: 2*n*(2*n-1)*a(n) = (43*n^2 - 53*n + 18)*a(n-1) - 3*(35*n^2 - 85*n + 54)*a(n-2) + (n-2)*(97*n - 165)*a(n-3) - 31*(n-3)*(n-2)*a(n-4). a(n) ~ 31^(n+1/2) / (9 * sqrt(Pi*n) * 2^(2*n+1)). (End) MAPLE a := n -> (hypergeom([1/3, 2/3, -n], [1/2, 1], -27/4) +2 ) / 3: seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 07 2015 MATHEMATICA FullSimplify[CoefficientList[Series[1 + x*D[Log[(2*Sin[(1/3)* ArcSin[(3/2)*Sqrt[3]* Sqrt[x/(1 - x)]]])/ (Sqrt[3]*Sqrt[(1 - x)* x])], x], {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 07 2015 *) Table[(1 + Sum[Binomial[n, j]*Binomial[3*j-1, j], {j, 0, n}])/2, {n, 0, 20}] (* Vaclav Kotesovec, Feb 07 2015 after Vladimir Kruchinin *) PROG (Maxima) a(n):=(1+sum(binomial(n, j)*binomial(3*j-1, j), j, 0, n))/2; (PARI) for(n=0, 25, print1((1 + sum(k=0, n, binomial(n, k)*binomial(3*k-1, k)))/2, ", ")) \\ G. C. Greubel, Jun 03 2017 CROSSREFS Cf. A165817, A188687. Sequence in context: A096656 A297014 A102009 * A298698 A135904 A145846 Adjacent sequences:  A254744 A254745 A254746 * A254748 A254749 A254750 KEYWORD nonn AUTHOR Vladimir Kruchinin, Feb 07 2015 STATUS approved

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Last modified September 17 10:52 EDT 2019. Contains 327129 sequences. (Running on oeis4.)