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
Keywords
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 and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
- Index entries for reversions of series
Programs
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Mathematica
CoefficientList[1/x InverseSeries[x*(1-x-x^3) + O[x]^26], x] (* Jean-François Alcover, Jul 20 2018 *)
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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 */
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PARI
Vec(serreverse(x*(1-x-x^3+O(x^66)))) /* Joerg Arndt, May 24 2011 */
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
Comments