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URL: https://oeis.org/A318614

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A318614
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).
0
1, 6, 76, 1260, 24276, 515592, 11721072, 280020312, 6945369860, 177358000248, 4635276570288, 123449340098448, 3339525750984528, 91535631253610400, 2537277723600799680, 71015600640006437040, 2004523477053308685540, 57003431104378084982040
OFFSET
1,2
COMMENTS
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].
REFERENCES
E. Heller, The Semiclassical Way to Dynamics and Spectroscopy, Princeton University Press, 2018, page 204.
LINKS
Eric Weisstein's World of Mathematics, Plane Division by Ellipses.
Eric Weisstein's World of Mathematics, Circle Ellipse Intersection.
FORMULA
(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(n) = A000108(n-1)*A098410(n-1).
EXAMPLE
Singular Value: S(1/2) = 1/sqrt(2).
N=4, h=1/sqrt(2) Quantization: S(u) = (n+1/2)*h/N.
n | u
==================================================
0 | 0.08544689553344134756293807606337...
1 | 0.23840989875904155311088418238272...
2 | 0.36638282702449450473835851051425...
3 | 0.46595506694324457665483887176081...
MATHEMATICA
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}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Bradley Klee, Aug 30 2018
STATUS
approved