OFFSET
0,3
COMMENTS
Conjecture: a(n) == binomial(4*n-1, n) (mod 2) for n >= 0 (cf. A263132).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..2000
EXAMPLE
G.f.: A(x) 1 + x + 3*x^2 + 9*x^3 + 25*x^4 + 60*x^5 + 111*x^6 + 356*x^7 + 717*x^8 + 1728*x^9 + 3532*x^10 + 7923*x^11 + 13947*x^12 + ...
RELATED SERIES.
1/A(x) = 1 - x - 2*x^2 - 4*x^3 - 6*x^4 + x^5 + 52*x^6 - 26*x^7 + 112*x^8 - 22*x^9 + 280*x^10 + 266*x^11 + 3978*x^12 + ...
The absolute value of the series 1/A(x) begins
abs(1/A(x)) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + x^5 + 52*x^6 + 26*x^7 + 112*x^8 + 22*x^9 + 280*x^10 + 266*x^11 + 3978*x^12 + 19476*x^13 + 53748*x^14 + 188096*x^15 + 356128*x^16 + 145318*x^17 + 4083268*x^18 + ...
the cube of which starts as
abs(1/A(x))^3 = 1 + 3*x + 9*x^2 + 25*x^3 + 60*x^4 + 111*x^5 + 356*x^6 + 717*x^7 + 1728*x^8 + ...
where A(x) = 1 + x*abs(1/A(x))^3.
SPECIFIC VALUES.
A(t) = 5 at t = 0.34652481192452632778148744009...
A(t) = 9/2 at t = 0.34332047911369115853530109434629340595421524344707...
A(t) = 4 at t = 0.33844988613244193281810217915341671138001247109315...
A(t) = 7/2 at t = 0.33093206633015553479076876378936259852312274709636...
A(t) = 3 at t = 0.31913094940940804614787566004609274666160372407803...
A(t) = 5/2 at t = 0.30017933266419626029599691715268323619028106096701...
A(t) = 2 at t = 0.26823879592468130644447947201722810537538246719689...
A(t) = 3/2 at t = 0.20641070526053514308343007863179336080812858639439...
A(1/3) = 3.6370099291721444216320225286542434877849899595617...
A(1/4) = 1.8094747379526694743161159394189701882898513040217...
A(1/5) = 1.4662568572713513624196239629654486684279393066965...
A(1/6) = 1.3230157298226165571635234305575666232122775793769...
A(1/7) = 1.2458642715965738773970674152984414596827918944570...
A(1/8) = 1.1980410385476832715212621689007173781378273728475...
PROG
(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + x*Ser(abs(Vec(1/(A +x*O(x^n)))))^3 ); polcoef(H=A, n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2025
STATUS
approved