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A217362 Series reversion of x + 2*x^2 + x^4. 1
1, -2, 8, -41, 236, -1456, 9412, -62922, 431464, -3017894, 21448024, -154438752, 1124289292, -8260992104, 61185802144, -456328537873, 3424063350108, -25830725790832, 195797025768044, -1490507771692940, 11390352356821680, -87348799726590300 (list; graph; refs; listen; history; text; internal format)
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

Apparently 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 = -0.0643333236178104225174767967838047014... - Vaclav Kotesovec, Sep 10 2013

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

Cf. A006013 (x - 2*x^2 + x^3).

Sequence in context: A060436 A020083 A333093 * A294084 A177340 A067119

Adjacent sequences:  A217359 A217360 A217361 * A217363 A217364 A217365

KEYWORD

sign

AUTHOR

R. J. Mathar, Oct 01 2012

STATUS

approved

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Last modified September 29 10:06 EDT 2020. Contains 337428 sequences. (Running on oeis4.)