A184994 E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! is inverse to f(x) = 2*sin(x) - x.

1, 2, 38, 2018, 210422, 36297362, 9356755718, 3369557048258, 1615758952865942, 995259055695876722, 765831994417031276198, 719917951968845731560098, 811830142106561351995390262, 1081642386040230828943441100882, 1680966987441826604383087455198278

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..100

Formula

a(n) = 2*((2*n)!*Sum_{k=1..(2*n)} binomial(2*n+k,2*n)*Sum_{j=1..k} binomial(k,j)*(Sum_{l=0..(j-1)} (binomial(j,l)*Sum_{i=0..(j-l)/2} binomial(j-l,i)*(l-j+2*i)^(2*n-l+j)*(-1)^(n-i)))/(2*n-l+j)!))))), a(0)=1.

a(n) ~ 2^(2*n+1) * n^(2*n) / (3^(1/4) * exp(2*n) * (sqrt(3) - Pi/3)^(2*n+1/2)). - Vaclav Kotesovec, Jan 26 2014

Mathematica

Table[(CoefficientList[InverseSeries[Series[-x+2*Sin[x], {x, 0, 31}], x], x]*Range[0, 31]!)[[n]], {n, 2, 30, 2}] (* Vaclav Kotesovec, Jan 26 2014 *)

Prog

(Maxima) a(n):=if n=0 then 1 else 2*((2*n)!*sum(binomial(2*n+k, 2*n)*sum(binomial(k, j)*(sum((binomial(j, l)*sum(binomial(j-l, i)*(l-j+2*i)^(2*n-l+j)*(-1)^(n-i), i, 0, (j-l)/2))/(2*n-l+j)!, l, 0, j-1)), j, 1, k), k, 1, 2*n));
(PARI) seq(n)={my(p=serlaplace(serreverse(2*sin(x + O(x^(2*n+2))) - x))); vector(n+1, i, polcoef(p, 2*i-1))} \\ Andrew Howroyd, Jan 04 2020

Crossrefs

Sequence in context: A187545 A246484 A288027 * A334554 A348162 A132396

Adjacent sequences: A184991 A184992 A184993 * A184995 A184996 A184997

Keyword

nonn

Author

Vladimir Kruchinin, Feb 04 2012

Extensions

Terms a(13) and beyond from Andrew Howroyd, Jan 04 2020

Status

approved