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A154825
Reversion of x*(1-2*x)/(1-3*x).
6
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, 206064020315, 572040039283, -590258340811
OFFSET
0,5
LINKS
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
KEYWORD
sign,easy
AUTHOR
Paul Barry, Jan 15 2009
STATUS
approved