A049171 Revert transform of 2*x*(1-x)-x/(1+x).

1, 1, 3, 9, 33, 125, 503, 2081, 8849, 38345, 168875, 753401, 3398177, 15469493, 70984559, 327982529, 1524644897, 7125440913, 33459931155, 157794990633, 747021246817, 3548843286829, 16912921740775, 80836929471329, 387397148131889, 1861088017162457

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

Sign diagram of generating sequence: +++-------------...

The sequences A049171 to A049189 are defined by series reversion of a sequence with rational (ordinary) generating function g(x). Solving g(x)=y for x yields algebraic equations for x, so the sequences have P-finite recurrences. - R. J. Mathar, Jul 24 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for reversions of series

Programs

Mathematica

Table[Sum[Binomial[3*k,k]*Binomial[n-1+k,3*k]/(2k+1)*2^k,{k,0,Floor[(n-1)/2]}],{n,1,20}] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n+1) = Sum_{k=0..floor(n/2)} A108759(n,k)*2^k. - Philippe Deléham, Dec 08 2009

Recurrence: 4*(n-1)*n*a(n) = 2*(n-1)*(5*n-6)*a(n-1) + 3*(16*n^2 - 67*n + 69)*a(n-2) + (25*n^2 - 169*n + 285)*a(n-3) + (n-4)*(2*n-9)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ sqrt(sqrt(3)-1)*((5+3*sqrt(3))/2)^n/(2*sqrt(6*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Extensions

NAME corrected by R. J. Mathar, Jul 23 2023