A217358 Series reversion of x-x^3-x^4.

1, 0, 1, 1, 3, 7, 16, 45, 110, 308, 819, 2275, 6328, 17748, 50388, 143412, 411939, 1187329, 3441559, 10015005, 29255655, 85766655, 252201690, 743819115, 2199446652, 6519727800, 19369551936, 57665571072, 172011364452, 514021640564, 1538650042952

Offset: 1

Author

R. J. Mathar, Oct 01 2012

Keywords

• nonn

Examples

			If y= x-x^3-x^4, then x= y + y^3 + y^4 +3*y^5 +7*y^6 +16*y^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A049140 (reversion of x-x^2-x^4).

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - x^3 - x^4, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Sep 10 2013 *)

Formula

Conjecture: 46*n*(n-1)*(n-2)*a(n) -(n-1)*(n-2)*(11*n-74)*a(n-1) -(n-2)*(336*n^2-1359*n+1351)*a(n-2) +(-347*n^3+2190*n^2-3861*n+1330)*a(n-3) + 8*(2*n-7)*(4*n-15)*(4*n-17)*a(n-4) = 0.

Recurrence (order 3): 23*(n-2)*(n-1)*n*(9*n-25)*a(n) = -(n-2)*(n-1)*(54*n^2 - 231*n + 248)*a(n-1) + (n-2)*(1485*n^3 - 10065*n^2 + 22292*n - 16088)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(9*n-16)*a(n-3). - Vaclav Kotesovec, Sep 10 2013

a(n) ~ c*d^n/n^(3/2), where d = 3/23*(2367+966*sqrt(3))^(1/3)+423/(23*(2367+966*sqrt(3))^(1/3))-2/23 = 3.145200906807902443... is the root of the equation -256 - 165*d + 6*d^2 + 23*d^3 = 0 and c = 1/48*sqrt(2)*sqrt((80793 + 65184*sqrt(3))^(1/3)*((80793 + 65184 * sqrt(3))^(2/3)-1839+9*(80793 + 65184 * sqrt(3))^(1/3)))/((80793 + 65184 * sqrt(3))^(1/3)*sqrt(Pi)) = 0.098446219937815765... - Vaclav Kotesovec, Sep 10 2013