A296726 -id:A296726

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A296727

Expansion of e.g.f. arcsinh(x)/(1 - x).

+10
2

0, 1, 2, 5, 20, 109, 654, 4353, 34824, 324441, 3244410, 34795485, 417545820, 5536151685, 77506123590, 1144330385625, 18309286170000, 315366695240625, 5676600514331250, 106667957800963125, 2133359156019262500, 45229212438054868125, 995042673637207098750, 22696937952367956440625

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

FORMULA

E.g.f.: log(x + sqrt(1 + x^2))/(1 - x).

a(n) ~ n! * log(1 + sqrt(2)). - Vaclav Kotesovec, Dec 20 2017

EXAMPLE

arcsinh(x)/(1 - x) = x/1! + 2*x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + ...

MAPLE

a:=series(arcsinh(x)/(1 - x), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 27 2019

MATHEMATICA

nmax = 23; CoefficientList[Series[ArcSinh[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

nmax = 23; CoefficientList[Series[Log[x + Sqrt[1 + x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

PROG

(PARI) first(n) = x='x+O('x^n); Vec(serlaplace(asinh(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017

CROSSREFS

Cf. A001818, A009628, A081358, A186763, A281964, A296726.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Dec 19 2017

STATUS

approved

A372324

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

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

Table of n, a(n) for n=0..22.

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

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

MATHEMATICA

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