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 A300058 a(n) = binomial(3*n,n)/(2*Pi)*Integral_{x=0..2*Pi} (12*cos^2(x)*sin(x) + 20*sin^3(x))^(2*n) dx. 3
 1, 492, 707220, 1298204880, 2654173160100, 5765723073622512, 13021894087331233104, 30217387890886676251200, 71532102917478013611243300, 171944976047709681477985038000, 418347201888204996027087975427920 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Compare with A295870. The series expansion "T(x)=2*Pi*sqrt(3/5)*Sum_{n>=0} a_n*(x/25)^n" determines the period T of anharmonic oscillation along a contour of the Hamiltonian energy surface "x=2H=(5/3)*p^2+q^2+4*(p^2+q^2)*q,0x'=1/108-x. A300057 has a similar definition to A005721, with a couple of extra integers appearing in the integrand. This makes a nice analogy between real and complex periods A295870, A300058. Second-order recurrences with polynomial coefficients define both sequences. LINKS Bradley Klee, Phase Plane Geometry. Brad Klee, Deriving Hypergeometric Picard-Fuchs Equations, Wolfram Demonstrations Project (2018). M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. FORMULA a(n) = A005809(n)*A300057(n). a(n) = Sum_{k1=0..2n} Sum_{k2=0..2n} binomial(3*n,n)*binomial(2*n,k2)*binomial(2*n,k1)*binomial(2*n,3*n-k1-k2)*((4+sqrt(15))^(2*n-k1))*((4-sqrt(15))^(2*n-k2)) a(0) = 1; a(1) = 492; a(n):=(c1/c3)*a(n-1)+(c2/c3)*a(n-2); with   c1 = 12*(-230+2259*n-3933*n^2+1863*n^3);   c2 = 5248800*(n-5/3)*(n-4/3)*(n-1/9);   c3 = 9*n^2*(n-10/9); a(n) ~ 2^(2*n - 1) * 3^(3*n - 1/2) * 5^(2*n + 1/2) / (Pi*n). - Vaclav Kotesovec, Apr 18 2018 MAPLE a := n -> 36^n*(3*n)!/n!^3*hypergeom([-2*n, n+1/2], [n+1], -2/3): seq(simplify(a(n)), n=0..10); # Peter Luschny, Apr 19 2018 MATHEMATICA c1=12*(-230+2259*n-3933*n^2+1863*n^3); c2=5248800*(n-5/3)*(n-4/3)*(n-1/9); c3=9*n^2*(n-10/9); a[0]=1; a[1]=492; a[n0_]:=ReplaceAll[(c1/c3)*a[n0-1]+(c2/c3)*a[n0-2], n->n0]; b[NN_]:=Expand[Total[Flatten[#]]&/@Table[Binomial[3*n, n]*Binomial[2*n, k2]*Binomial[2*n, k1]*Binomial[2*n, 3*n-k1-k2]*((4+Sqrt[15])^(2*n-k1))*((4-Sqrt[15])^(2*n-k2)), {n, 0, NN}, {k1, 0, 2*n}, {k2, 0, 2*n}]]; ({#, SameQ[#, a/@Range[0, 10]]}&@b[10])[[1]] Table[Binomial[3*n, n] * SeriesCoefficient[(1 + 9*z + 9*z^2 + z^3)^(2*n), {z, 0, 3*n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 18 2018 *) CROSSREFS Cf. A002894, A113424, A006480, A000897. Factors: A005809, A300057. Real Period: A295870. Sequence in context: A231455 A007242 A198531 * A251540 A116207 A264327 Adjacent sequences:  A300055 A300056 A300057 * A300059 A300060 A300061 KEYWORD nonn AUTHOR Bradley Klee, Feb 23 2018 STATUS approved

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Last modified January 23 16:36 EST 2020. Contains 331172 sequences. (Running on oeis4.)