A372324 Expansion of e.g.f. arcsin(x)^2/(2*(1 - x)).

0, 0, 1, 3, 16, 80, 544, 3808, 32768, 294912, 3096576, 34062336, 423493632, 5505417216, 79199207424, 1187988111360, 19423989596160, 330207823134720, 6050282848911360, 114955374129315840, 2333627101111910400, 49006169123350118400, 1091943568123940044800

Offset: 0

Author

Greg Dresden, Apr 27 2024

Keywords

• nonn

Comments

a(2*n) appears in the formula for the limit, as k -> infinity, of the area between cos^(2*n)(x) and cos^(2*n)(k*x) on the interval [0, Pi]. To be precise, here is the formula: a(2*n)*(16/Pi)/((2*n)!!)^2 = Lim_{k->oo} Integral_{x=0..Pi} abs(cos^(2*n)(x) - cos^(2*n)(k*x)) dx. See the article by Dombrowski and Dresden.

Links

• Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024.

Crossrefs

Cf. A296726.

Programs

Mathematica

Table[n! SeriesCoefficient[ArcSin[x]^2/(2 (1 - x)), {x, 0, n}], {n, 0, 22}]

Formula

a(2*n+1) = (2*n+1)*a(2*n).

a(2*n) = (2*n)*(2*n-1)*a(2*n-2) + ((2*n-2)!!)^2.

a(n) = (n!)*Sum_{k=0..(n-2)/2} ((2*k)!!)/(((2*k+1)!!)*(2*k+2)).

E.g.f.: arcsin(x)^2/(2*(1 - x)).

a(n) ~ n! * (Pi^2/8) * (1 - 2^(5/2)/(Pi^(3/2)*sqrt(n))). - Vaclav Kotesovec, May 01 2024

D-finite with recurrence a(n) -n*a(n-1) -(n-2)^2*a(n-2) +(n-2)^3*a(n-3)=0. - R. J. Mathar, May 02 2024