A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).
0, 1, -2, 18, -312, 9470, -436860, 28616322, -2522596496, 288046961190, -41355026494020, 7291524732108650, -1548849359704927896, 390122366308850972238, -114968364853645904762252, 39189956630839558368115410, -15300235972710835734174638880
Offset: 0
Keywords
Examples
log(1 + x*tan(x/2)) = x^2/2! - 2*x^4/4! + 18*x^6/6! - 312*x^8/8! + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..200
Programs
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Mathematica
nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
Formula
a(n) = (2*n)! * [x^(2*n)] log(1 + x*tan(x/2)).
a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.54340463841820844795870974005331555369788376471926269... is the root of the equation r*tanh(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017