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A380709 G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^3.

%I A380709 #7 Feb 10 2025 04:39:25
%S A380709 1,1,3,9,25,60,111,356,717,1728,3532,7923,13947,43956,135762,455844,
%T A380709 1502005,4377084,9696816,33777040,76261380,211981800,491690441,
%U A380709 1156806114,2388107247,7425085120,22208783472,72885740508,243066599038,726160343256,1695120635568,5836780502656,13416367141485
%N A380709 G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^3.
%C A380709 Conjecture: a(n) == binomial(4*n-1, n) (mod 2) for n >= 0 (cf. A263132).
%H A380709 Paul D. Hanna, <a href="/A380709/b380709.txt">Table of n, a(n) for n = 0..2000</a>
%e A380709 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 + ...
%e A380709 RELATED SERIES.
%e A380709 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 + ...
%e A380709 The absolute value of the series 1/A(x) begins
%e A380709 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 + ...
%e A380709 the cube of which starts as
%e A380709 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 + ...
%e A380709 where A(x) = 1 + x*abs(1/A(x))^3.
%e A380709 SPECIFIC VALUES.
%e A380709 A(t) = 5   at t = 0.34652481192452632778148744009...
%e A380709 A(t) = 9/2 at t = 0.34332047911369115853530109434629340595421524344707...
%e A380709 A(t) = 4   at t = 0.33844988613244193281810217915341671138001247109315...
%e A380709 A(t) = 7/2 at t = 0.33093206633015553479076876378936259852312274709636...
%e A380709 A(t) = 3   at t = 0.31913094940940804614787566004609274666160372407803...
%e A380709 A(t) = 5/2 at t = 0.30017933266419626029599691715268323619028106096701...
%e A380709 A(t) = 2   at t = 0.26823879592468130644447947201722810537538246719689...
%e A380709 A(t) = 3/2 at t = 0.20641070526053514308343007863179336080812858639439...
%e A380709 A(1/3) = 3.6370099291721444216320225286542434877849899595617...
%e A380709 A(1/4) = 1.8094747379526694743161159394189701882898513040217...
%e A380709 A(1/5) = 1.4662568572713513624196239629654486684279393066965...
%e A380709 A(1/6) = 1.3230157298226165571635234305575666232122775793769...
%e A380709 A(1/7) = 1.2458642715965738773970674152984414596827918944570...
%e A380709 A(1/8) = 1.1980410385476832715212621689007173781378273728475...
%o A380709 (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)}
%o A380709 for(n=0,40,print1(a(n),", "))
%Y A380709 Cf. A263132, A380708.
%K A380709 nonn
%O A380709 0,3
%A A380709 _Paul D. Hanna_, Feb 09 2025