A063033 Reversion of y - y^2 + y^4.
0, 1, 1, 2, 4, 8, 14, 16, -21, -242, -1166, -4472, -15132, -46508, -130016, -323000, -660535, -786714, 1789952, 18546464, 93845290, 380532240, 1355983860, 4363436280, 12688926510, 32530717752, 67666586472, 76255301640, -240266135872
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..800
- Elżbieta Liszewska, 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[InverseSeries[Series[y - y^2 + y^4, {y, 0, 30}], x], x] -
Maxima
a(n):=sum((-1)^j*binomial(n-2*j-1,j)*binomial(2*n-2*j-2,n-1),j,0,(n-1)/2)/n; /* Vladimir Kruchinin, Oct 11 2011 */
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PARI
concat(0, Vec(serreverse(y - y^2 + y^4 + O(y^10)))) \\ Michel Marcus, Jun 28 2018
Formula
a(n) = Sum_{j=0..(n-1)/2} (-1)^j*binomial(n-2*j-1, j)*binomial(2*n-2*j-2, n-1)/n, a(0)=0. - Vladimir Kruchinin, Oct 11 2011
D-finite with recurrence 391*n*(n-1)*(n-2)*a(n) -8*(n-1)*(n-2)*(203*n -132)*a(n-1) -4*(n-2)*(224*n^2 -2816*n +5697)*a(n-2) +8*(928*n^3 -7920*n^2 +22682*n-21915)*a(n-3) +192*(4*n-15) *(2*n-7)*(4*n-17)*a(n-4)=0, n-4>=1 - R. J. Mathar, Mar 24 2023