A063028 - OEIS

%I #36 Mar 21 2022 06:30:12

%S 0,1,1,2,5,13,35,96,264,720,1925,4966,12038,25907,41310,-5168,-468996,

%T -2982240,-14350320,-61334790,-244951840,-934684465,-3447083370,

%U -12365767620,-43304717625,-148314737961,-497033803314,-1628721662260,-5208556347700

%N Reversion of x - x^2 + x^5.

%C For the reversion of x - a*x^2 - b*x^5 (a!=0, b!=0) we have a(n) = Sum_{j=0..floor((n-1)/3)} a^(n-4*j-1)*b^j*binomial(n-3*j-1, j)*binomial(2*n-3*j-2, n-1)/n, n > 0. - _Vladimir Kruchinin_, May 28 2011

%H Vincenzo Librandi, <a href="/A063028/b063028.txt">Table of n, a(n) for n = 0..125</a>

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n) = Sum_{j=0..floor((n-1)/3)} (-1)^j*binomial(n-3*j-1, j)*binomial(2*n-3*j-2, n-1)/n, n > 0, a(0)=0. - _Vladimir Kruchinin_, May 28 2011

%F D-finite with recurrence +7576007*n*(n-1)*(n-2)*(n-3)*a(n) -14*(n-1)*(n-2)*(n-3)*(2435499*n+162464)*a(n-1) -70*(n-2)*(n-3)*(443090*n^2-7345575*n+16893064)*a(n-2) +1500*(n-3)*(126085*n^3-1478320*n^2+5789009*n-7573118)*a(n-3) +5*(8864375*n^4-23685000*n^3-505107125*n^2+2934387750*n-4417359408)*a(n-4) +250*(5*n-26)*(166625*n^3-1966175*n^2+7613615*n-9631377)*a(n-5) -131250*(5*n-27)*(5*n-31)*(5*n-24)*(5*n-28)*a(n-6)=0. - _R. J. Mathar_, Mar 21 2022

%t CoefficientList[InverseSeries[Series[y - y^2 + y^5, {y, 0, 30}], x], x]

%o (Maxima)

%o a(n):=sum((-1)^j*binomial(n-3*j-1,j)*binomial(2*n-3*j-2,n-1),j,0,(n-1)/3)/n; /* _Vladimir Kruchinin_, May 28 2011 */

%o (PARI) x='x+O('x^66); /* that many terms */

%o Vec(serreverse(x-x^2+x^5)) /* show terms */ /* _Joerg Arndt_, May 28 2011 */

%K sign,easy

%O 0,4

%A _Olivier Gérard_, Jul 05 2001