login
Series reversion of x+x^3+x^4.
1

%I #11 Mar 21 2022 06:26:35

%S 1,0,-1,-1,3,7,-8,-45,0,264,273,-1365,-3192,5508,27132,-7752,-193743,

%T -158631,1177209,2417415,-5673525,-23595585,14488110,187050435,

%U 104481780,-1251127512,-2178989008,6775504088,23824892148,-23395134188,-204487059656,-57418615353,1471227866951

%N Series reversion of x+x^3+x^4.

%H R. J. Mathar, <a href="/A217359/b217359.txt">Table of n, a(n) for n = 1..104</a>

%F D-finite with recurrence 124*n*(n-1)*(n-2)*a(n) +(n-1)*(n-2)*(7*n-88)*a(n-1) +(n-2)*(870*n^2-3465*n+3347)*a(n-2) +(1243*n^3-9870*n^2+25869*n-22490)*a(n-3) +8*(4*n-15)*(2*n-7)*(4*n-17)*a(n-4) = 0.

%F Recurrence (order 3): 31*(n-2)*(n-1)*n*(15*n-41)*a(n) = (n-2)*(n-1)*(90*n^2 - 381*n + 400)*a(n-1) - (n-2)*(3285*n^3 - 22119*n^2 + 48706*n - 34960)*a(n-2) - 8*(2*n-5)*(4*n-13)*(4*n-11)*(15*n-26)*a(n-3). - _Vaclav Kotesovec_, Sep 10 2013

%F Lim sup n->infinity |a(n)|^(1/n) = 16/sqrt(31) = 2.8736848... - _Vaclav Kotesovec_, Sep 10 2013

%e If y= x+x^3+x^4, then x=y -y^3 -y^4 +3*y^5 +7*y^6 -8*y^7 -45*y^8 +...

%t Rest[CoefficientList[InverseSeries[Series[x+x^3+x^4,{x,0,20}],x],x]] (* _Vaclav Kotesovec_, Sep 10 2013 *)

%Y Cf. A217358 (x-x^3-x^4).

%K sign,easy

%O 1,5

%A _R. J. Mathar_, Oct 01 2012