A049140 Revert transform of 1 - x - x^3.

1, 1, 2, 6, 20, 70, 256, 969, 3762, 14894, 59904, 244088, 1005452, 4180096, 17516936, 73913705, 313774854, 1339162028, 5742691704, 24731501410, 106919054880, 463844340060, 2018673093000, 8810852089650, 38558866555248

Offset

1,3

Comments

Series reversion of x-x^2-x^4. - Joerg Arndt, May 24 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..100, format errors corrected by Vaclav Kotesovec, Aug 07 2013

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 659

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Index entries for reversions of series

Formula

a(n) = sum(j=0..(n-1)/2, binomial(n-2*j-1,j)*binomial(2*n-2*j-2,n-1))/n. - Vladimir Kruchinin, May 24 2011

D-finite with recurrence 31*n*(n-1)*(n-2)*(140*n-383)*a(n) -8*(n-1)*(n-2)*(2800*n^2 -11860*n+11583)*a(n-1) +4*(n-2)*(4480*n^3-30176*n^2+66916*n-48753)*a(n-2) -8*(4*n-11)*(4*n-13)*(140*n-243)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Sep 29 2012

Mathematica

CoefficientList[1/x InverseSeries[x*(1-x-x^3) + O[x]^26], x] (* Jean-François Alcover, Jul 20 2018 *)

Prog

(Maxima)
a(n):=sum(binomial(n-2*j-1, j)*binomial(2*n-2*j-2, n-1), j, 0, (n-1)/2)/n; /* Vladimir Kruchinin, May 24 2011 */
(PARI) Vec(serreverse(x*(1-x-x^3+O(x^66)))) /* Joerg Arndt, May 24 2011 */

Crossrefs

Sequence in context: A369630 A185202 A340891 * A092413 A151285 A150126

Adjacent sequences: A049137 A049138 A049139 * A049141 A049142 A049143

Keyword

nonn

Author

Olivier Gérard

Status

approved