A063023 - OEIS

%I #37 Mar 24 2023 15:20:45

%S 0,1,1,2,6,21,77,292,1143,4592,18821,78364,330512,1409149,6063526,

%T 26298592,114849110,504595293,2228824203,9891723114,44087704836,

%U 197255893945,885630834120,3988872011820,18017892014655

%N Reversion of y - y^2 - y^4 - y^5.

%H Robert Israel, <a href="/A063023/b063023.txt">Table of n, a(n) for n = 0..1471</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(k=0..n-1, (sum(j=floor((n-k-1)/3)..floor((n-k-1)/2), binomial(j,n-k-2*j-1)*binomial(k,j)))*binomial(n+k-1,n-1))/n, n>0, a(0)=0. - _Vladimir Kruchinin_, May 26 2011

%F D-finite with recurrence 500*n*(n-1)*(n-2)*(n-3)*(2749734646741*n -12465511712206)*a(n) -4350*(n-1)*(n-2)*(n-3)*(1671932639004*n^2 -9463993359665*n +8542758180998)*a(n-1) +6*(n-2)*(n-3)*(920566361953318*n^3 -5234275380165135*n^2 +1599947907526062*n + 14550951937253720)*a(n-2) -58*(n-3)*(141125952976394*n^4 -1372074713192783*n^3 +3865929553111591*n^2 -1368848933612182*n -5017756051156800)*a(n-3) +3*(-5087969954630151*n^5 +92718860230184360*n^4 -673277365411431865*n^3 +2437688480550464340*n^2 -4405429911017929324*n +3182000337895328880)*a(n-4) -145*(5*n-22)*(5*n-23)*(5*n-26)*(48367137647*n -140254765035)*(5*n-24)*a(n-5)=0. - _R. J. Mathar_, Mar 21 2022

%p with(gfun):

%p F:= RootOf(y-y^2-y^4-y^5-x, y):

%p DE:=holexprtodiffeq(F, g(x)):

%p Rec:= diffeqtorec(DE, g(x), a(n)):

%p f:= rectoproc(Rec, a(n), remember):

%p map(f, [$0..50]);# _Robert Israel_, Jan 08 2019

%t CoefficientList[InverseSeries[Series[y - y^2 - y^4 - y^5, {y, 0, 30}], x], x]

%o (Maxima)

%o a(n):=sum((sum(binomial(j,n-k-2*j-1)*binomial(k,j),j,floor((n-k-1)/3),floor((n-k-1)/2)))*binomial(n+k-1,n-1),k,0,n-1)/n; /* _Vladimir Kruchinin_, May 26 2011 */

%o (Sage) # uses[Reversion from A063022]

%o Reversion(x - x^2 - x^4 - x^5, 25) # _Peter Luschny_, Jan 08 2019

%o (PARI) concat(0, Vec(serreverse(x - x^2 - x^4 - x^5 + O(x^30)))) \\ _Michel Marcus_, Jan 08 2019

%Y Cf. A063019, A063022.

%K nonn,easy

%O 0,4

%A _Olivier Gérard_, Jul 05 2001