A013463 Expansion of e.g.f. sin(arctan(x) - arctanh(x)) (odd powers only).

0, -4, 0, -1440, 17920, -7257600, 395366400, -175791616000, 24049778688000, -13090802909184000, 3482386518507520000, -2338795470534082560000, 1043344639170183168000000, -855872901958901432320000000, 581497055554547881082880000000, -574895744297242087233945600000000

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..224

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] sin(arctan(x)-arctanh(x)). - Alois P. Heinz, Aug 20 2014

16*(2*k + 7)*(2*k + 5)*(2*k + 3)*(2*k + 1)*(k + 4)*(k + 2)*(k + 1)^2*a(k) + 16*(k + 2)*(2*k + 5)*(2*k + 7)*(k + 4)*a(k+1) - 8*(2*k + 7)*(2*k + 5)*(k + 2)*(k + 4)*a(k+2) + a(k+4) = 0. - Robert Israel, Dec 18 2018

a(n) = (2*n+1)! * [x^n] (((1-sqrt(x))/(1+sqrt(x)))^(i/2)*(sqrt(x)-i) + ((1+sqrt(x))/(1-sqrt(x)))^(i/2)*(sqrt(x)+i)) / (2*sqrt(x*(x+1))). - Peter Luschny, Dec 30 2025

Example

sin(arctan(x) - arctanh(x)) = -4/3!*x^3 -1440/7!*x^7 +17920/9!*x^9 ...

Maple

f:= sin(arctan(x)-arctanh(x)):
S:= series(f, x, 62):
seq((2*k+1)!*coeff(S, x, 2*k+1), k=0..30); # Robert Israel, Dec 18 2018
a := proc(n) simplify( (-1/4)^n*binomial(2*n, n)*(2*n+1)!*((2*n+1)*(-hypergeom([1/4*I, -n-1], [1/2], 2)+hypergeom([-n, 1/4*I], [1/2], 2))*hypergeom([-1/4*I, -n-1], [1/2], 2)-hypergeom([-n, -1/4*I], [1/2], 2)*(2*hypergeom([-n, 1/4*I], [1/2], 2)*n-2*hypergeom([1/4*I, -n-1], [1/2], 2)*n-hypergeom([1/4*I, -n-1], [1/2], 2))) ) end:
seq(a(n), n=0..20); # Mark van Hoeij, Dec 30 2025
f := ln((1-sqrt(x))/(1+sqrt(x)))/2: egf := (sqrt(x)*cos(f) + sin(f))/sqrt(x*(x+1)):
ser := series(egf, x, 16): seq((2*n+1)!*coeff(ser, x, n), n = 0..15);  # Peter Luschny, Dec 30 2025

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Sin[ArcTan[x]-ArcTanh[x]], {x, 0, nn}], x] Range[0, nn-1]!, {2, -1, 2}]] (* Harvey P. Dale, Aug 14 2014 *)
gf := (((1-Sqrt[x])/(1+Sqrt[x]))^(I/2)*(Sqrt[x]-I)  + ((1+Sqrt[x])/(1-Sqrt[x]))^(I/2)*(Sqrt[x]+I)) / (2*Sqrt[x*(x+1)]); coeffs = CoefficientList[Series[gf, {x, 0, 15}], x];
MapIndexed[#1 * Factorial[2*(#2[[1]] - 1) + 1] &, coeffs] (* Peter Luschny, Dec 30 2025 *)

Crossrefs

Sequence in context: A216675 A012502 A130105 * A013464 A331582 A306819

Adjacent sequences: A013460 A013461 A013462 * A013464 A013465 A013466

Keyword

sign

Author

Patrick Demichel

Extensions

Definition clarified by Harvey P. Dale, Aug 14 2014

a(0)=0 prepended by Joerg Arndt, Aug 19 2014

Status

approved