A154825 Reversion of x*(1-2*x)/(1-3*x).

1, -1, -1, 1, 5, 3, -21, -51, 41, 391, 407, -1927, -6227, 2507, 49347, 71109, -236079, -966129, 9519, 7408497, 13685205, -32079981, -167077221, -60639939, 1209248505, 2761755543, -4457338681, -30629783831, -22124857219

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: (1+3*x-sqrt(1-2*x+9*x^2))/(4*x). - corrected by Vaclav Kotesovec, Feb 08 2014

G.f.: 1/(1+x/(1-2x/(1+x/(1-2x/(1+x/(1-2x/(1+.... (continued fraction).

a(n) = Sum_{k=0..n} binomial(n+k, 2k)*A000108(k)*2^k*(-3)^(n-k).

From Philippe Deléham, Jan 17 2009: (Start)

a(n) = Sum_{k=0..n} A131198(n,k)*(-1)^(n-k)*2^k.

a(n) = Sum_{k=0..n} A090181(n,k)*(-1)^k*2^(n-k).

a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k)*(-3)^k.

a(n) = Sum_{k=0..n} A088617(n,k)*2^k*(-3)^(n-k).

a(n) = Sum_{k=0..n} A086810(n,k)*(-1)^k*3^(n-k).

a(n) = Sum_{k=0..n} A133336(n,k)*3^k*(-1)^((n-k). (End)

D-finite with recurrence (n+1)*a(n) = (2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - R. J. Mathar, Nov 15 2012

a(n) = (-3)^n*Hypergeometric2F1([-n, n+1], [2]; 2/3). - G. C. Greubel, May 24 2022

Maple

A154825_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := -a[w-1]+2*add(a[j]*a[w-j-1], j=1..w-1)od;
convert(a, list) end: A154825_list(28); # Peter Luschny, May 19 2011

Mathematica

CoefficientList[Series[(1+3*x-Sqrt[1-2*x+9*x^2])/(4*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1+3*x-Sqrt(1-2*x+9*x^2))/(4*x) )); // G. C. Greubel, May 24 2022
(SageMath) [sum(binomial(n+k, n-k)*catalan_number(k)*2^k*(-3)^(n-k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, May 24 2022

Crossrefs

Cf. A000108, A060693, A086810, A088617, A090181, A091593, A131198, A133336.

Sequence in context: A221473 A199638 A296356 * A091593 A139699 A303634

Adjacent sequences: A154822 A154823 A154824 * A154826 A154827 A154828

Keyword

sign,easy

Author

Paul Barry, Jan 15 2009

Status

approved