A379606 S = (A-sin(A))/2 gives the area of a segment of the unit circle in terms of the arc length A (<= Pi). Expressing A in terms of S we get A = Sum_{n>=0} b^(2n+1)*c(n) where b = (12*S)^(1/3). Sequence gives numerators of c(n).
1, 1, 1, 1, 43, 1213, 151439, 33227, 16542537833, 887278009, 15233801224559, 9597171184603, 1374085664813273149, 1593410154419351, 53299328587804322691259, 1065024810026227256263721, 11374760871959174491194191, 70563256104582737796094772987, 657272463951301325116190773432261
Offset: 0
Examples
A = b + b^3/60 + b^5/1400 + b^7/25200 + ..., where b = (12*S)^(1/3); the c(n) are 1, 1/60, 1/1400, 1/25200, 43/17248000, 1213/7207200000, ...
Crossrefs
Cf. A379607 (denominators).
Programs
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Mathematica
Numerator[CoefficientList[InverseSeries[Series[Surd[(6*(x - Sin[x])), 3], {x, 0, 40}]], x][[2 ;; -2 ;; 2]]] (* Amiram Eldar, Dec 27 2024 *)
Extensions
Edited by N. J. A. Sloane, Jan 14 2025