OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
R. J. Mathar, Series Expansion of Generalized Fresnel Integrals, arXiv:1211.3963 [math.CA], 2012.
FORMULA
D-finite with recurrence 6313*n*(n-1)*(n-2)*a(n) +16*(n-1)*(n-2)*(3461*n-4749)*a(n-1) +16*(n-2)*(1792*n^2-3088*n-3267)*a(n-2) +8*(352*n^3+5040*n^2-40030*n+64755)*a(n-3) -192*(4*n-15)*(4*n-14)*(4*n-17)*a(n-4)=0.
Recurrence (order 3): 59*(n-2)*(n-1)*n*(364*n-985)*a(n) = -32*(n-2)*(n-1)*(6188*n^2 - 26027*n + 25206)*a(n-1) - 16*(n-2)*(11648*n^3 - 78112*n^2 + 172196*n - 124479)*a(n-2) - 8*(2*n-5)*(4*n-13)*(4*n-11)*(364*n-621)*a(n-3). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = -4/177*(1547974 + 32214 * sqrt(273))^(1/3) - 51328/(177*(1547974 + 32214 * sqrt(273))^(1/3))-544/177 = -8.229964287393075... is the root of the equation 256 + 512*d + 544*d^2 + 59*d^3 = 0 and c = -sqrt(cosh(arccosh(2281*sqrt(91)/2048)/3) / (3*sqrt(91)) - 1/24)/sqrt(Pi) = -0.0643333236178104225174767967838047014... - Vaclav Kotesovec, Sep 10 2013, updated Mar 17 2024
a(n) = ((-1)^(n-1)*sum(j=0..(n-1)/2), 2^(n-3*j-1)*binomial(n-2*j-1,j)*binomial(2*n-2*j-2,n-1))/n. - Vladimir Kruchinin, May 17 2014
a(n) = ((-8)^(n-1)*Gamma(n-1/2))/(sqrt(Pi)*Gamma(n+1))*hypergeom([(1-n)/3, (2-n)/3, (3-n)/3], [1-n, 3/2-n], -27/32) for n>1. - Peter Luschny, May 18 2014
EXAMPLE
If y = x + 2x^2 + x^4, then x = y - 2*y^2 + 8*y^3 - 41*y^4 + 236*y^5 - ...
MAPLE
a := n -> ((-8)^(n-1)*GAMMA(n-1/2))/(sqrt(Pi)*GAMMA(n+1))* hypergeom([(1-n)/3, (2-n)/3, (3-n)/3], [1-n, 3/2-n], -27/32); seq(round(evalf(a(n), 32)), n=1..22); # Peter Luschny, May 18 2014
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x+2*x^2+x^4, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Sep 10 2013 *)
PROG
(Maxima)
a(n):=((-1)^(n-1)*sum(2^(n-3*j-1)*binomial(n-2*j-1, j)*binomial(2*n-2*j-2, n-1), j, 0, (n-1)/2))/n; /* Vladimir Kruchinin, May 17 2014 */
(PARI) for(n=1, 50, print1((-1)^(n-1)*sum(j=0, (n-1)/2, 2^(n-3*j-1)*binomial(n-2*j-1, j)*binomial(2*n-2*j-2, n-1))/n, ", " )) \\ G. C. Greubel, Apr 01 2017
CROSSREFS
KEYWORD
sign,easy
AUTHOR
R. J. Mathar, Oct 01 2012
STATUS
approved