A318614 - OEIS

%I #18 Jan 26 2024 15:49:44

%S 1,6,76,1260,24276,515592,11721072,280020312,6945369860,177358000248,

%T 4635276570288,123449340098448,3339525750984528,91535631253610400,

%U 2537277723600799680,71015600640006437040,2004523477053308685540,57003431104378084982040

%N Scaled g.f. S(u) = Sum_{n>0} a(n)*16*(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).

%C Area interior to the central loop of u = 2*H = x^2 + y^2 - (1/2)*(x^4 + y^4) equals to Pi*S(u), when u in [0,1/2].

%D E. Heller, The Semiclassical Way to Dynamics and Spectroscopy, Princeton University Press, 2018, page 204.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlaneDivisionbyEllipses.html">Plane Division by Ellipses</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Circle-EllipseIntersection.html">Circle Ellipse Intersection</a>.

%F (n-1)^2*n*a(n) - 12*(n-1)*(2*n-3)^2*a(n-1) + 128*(n-2)*(2*n-5)*(2*n-3)*a(n-2) == 0.

%F a(n) = A000108(n-1)*A098410(n-1).

%e Singular Value: S(1/2) = 1/sqrt(2).

%e N=4, h=1/sqrt(2) Quantization: S(u) = (n+1/2)*h/N.

%e   n  |                  u

%e ==================================================

%e   0  |  0.08544689553344134756293807606337...

%e   1  |  0.23840989875904155311088418238272...

%e   2  |  0.36638282702449450473835851051425...

%e   3  |  0.46595506694324457665483887176081...

%t RecurrenceTable[{(n-1)^2*n*a[n] - 12*(n-1)*(2*n-3)^2*a[n-1] + 128*(n-2)*(2*n-5)*(2*n-3)*a[n-2] == 0, a[1] == 1, a[2] == 6}, a, {n, 1, 1000}]

%o (GAP) a:=[1,6];; for n in [3..20] do a[n]:=(1/(n*(n-1)^2))*(12*(n-1)*(2*n-3)^2*a[n-1]-(128*(n-2)*(2*n-5)*(2*n-3)*a[n-2])); od; a; # _Muniru A Asiru_, Sep 24 2018

%Y Cf. A318417, A010503, A247719, A000888.

%K nonn

%O 1,2

%A _Bradley Klee_, Aug 30 2018