# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a300058 Showing 1-1 of 1 %I A300058 #33 Sep 12 2018 02:18:10 %S A300058 1,492,707220,1298204880,2654173160100,5765723073622512, %T A300058 13021894087331233104,30217387890886676251200, %U A300058 71532102917478013611243300,171944976047709681477985038000,418347201888204996027087975427920 %N 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. %C A300058 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. %C A300058 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. %H A300058 Bradley Klee, Phase Plane Geometry. %H A300058 Brad Klee, Deriving Hypergeometric Picard-Fuchs Equations, Wolfram Demonstrations Project (2018). %H A300058 M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. %F A300058 a(n) = A005809(n)*A300057(n). %F A300058 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)) %F A300058 a(0) = 1; a(1) = 492; a(n):=(c1/c3)*a(n-1)+(c2/c3)*a(n-2); with %F A300058 c1 = 12*(-230+2259*n-3933*n^2+1863*n^3); %F A300058 c2 = 5248800*(n-5/3)*(n-4/3)*(n-1/9); %F A300058 c3 = 9*n^2*(n-10/9); %F A300058 a(n) ~ 2^(2*n - 1) * 3^(3*n - 1/2) * 5^(2*n + 1/2) / (Pi*n). - _Vaclav Kotesovec_, Apr 18 2018 %p A300058 a := n -> 36^n*(3*n)!/n!^3*hypergeom([-2*n, n+1/2], [n+1], -2/3): %p A300058 seq(simplify(a(n)), n=0..10); # _Peter Luschny_, Apr 19 2018 %t A300058 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]; %t A300058 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]] %t A300058 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 *) %Y A300058 Cf. A002894, A113424, A006480, A000897. Factors: A005809, A300057. Real Period: A295870. %K A300058 nonn %O A300058 0,2 %A A300058 _Bradley Klee_, Feb 23 2018 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE