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A386659 G.f. A(x) satisfies A(x^3) = A(x)^3/(1 + 3*A(x)).

%I A386659 #18 Aug 30 2025 15:56:47
%S A386659 1,1,0,0,1,0,-1,1,0,-2,3,0,-6,10,0,-22,33,0,-79,122,0,-299,472,0,
%T A386659 -1179,1871,0,-4754,7601,0,-19553,31449,0,-81720,132020,0,-345949,
%U A386659 561034,0,-1480475,2408712,0,-6394189,10431950,0,-27835400,45521500,0,-122008360,199948108,0,-538016031,883331845,0
%N A386659 G.f. A(x) satisfies A(x^3) = A(x)^3/(1 + 3*A(x)).
%C A386659 Compare to: F(x^2) = F(x)^2/(1 + 2*F(x)) holds when F(x) = x/(1-x).
%H A386659 Paul D. Hanna, <a href="/A386659/b386659.txt">Table of n, a(n) for n = 1..2200</a>
%F A386659 G.f. A(x) = Sum_{n>=1} a(n)*x^n along with series trisections T1(x) = Sum_{n>=0} a(3*n+1)*x^(3*n+1) and T2(x) = Sum_{n>=0} a(3*n+2)*x^(3*n+2) satisfy the following formulas.
%F A386659 (1) A(x^3) = A(x)^3/(1 + 3*A(x)).
%F A386659 (2) a(3*n) = 0 for n >= 1.
%F A386659 (3) T1(x)*T2(x) = A(x^3).
%F A386659 (4) T2(x)/T1(x) = G(x^3)/x^2 where g.f. G(x) of A370446 satisfies G(x)^3 + x^4/G(x)^3 = G(x^3) + x^4/G(x^3) - 3*x^2.
%F A386659 (5) A(-F(-x)) = x where g.f. F(x) of A264228 satisfies F(x)^3 = F( x^3/(1-3*x) ).
%e A386659 G.f.: A(x) = x + x^2 + x^5 - x^7 + x^8 - 2*x^10 + 3*x^11 - 6*x^13 + 10*x^14 - 22*x^16 + 33*x^17 - 79*x^19 + 122*x^20 - 299*x^22 + 472*x^23 - 1179*x^25 + 1871*x^26 - 4754*x^28 + ...
%e A386659 where A(x^3) = A(x)^3/(1 + 3*A(x)).
%e A386659 RELATED SERIES.
%e A386659 The series trisections are A(x) = T1(x) + T2(x) + T3(x), with T3(x) = 0 and
%e A386659 T1(x) = x - x^7 - 2*x^10 - 6*x^13 - 22*x^16 - 79*x^19 - 299*x^22 - 1179*x^25 - 4754*x^28 - 19553*x^31 - 81720*x^34 - 345949*x^37 - 1480475*x^40 + ...
%e A386659 T2(x) = x^2 + x^5 + x^8 + 3*x^11 + 10*x^14 + 33*x^17 + 122*x^20 + 472*x^23 + 1871*x^26 + 7601*x^29 + 31449*x^32 + 132020*x^35 + 561034*x^38 + 2408712*x^41 + ...
%e A386659 where T1(x)*T2(x) = A(x^3) and
%e A386659 T2(x)/T1(x) = x + x^4 + 2*x^7 + 6*x^10 + 20*x^13 + 71*x^16 + 267*x^19 + 1041*x^22 + 4168*x^25 + 17047*x^28 + ... + A370446(n)*x^(3*n-2) + ...
%e A386659 The cube of A(x) also has interesting series trisections.
%e A386659 A(x)^3 = x^3 + 3*x^4 + 3*x^5 + x^6 + 3*x^7 + 6*x^8 - 3*x^10 + 6*x^11 - 9*x^13 + 12*x^14 + x^15 - 21*x^16 + 42*x^17 - 84*x^19 + 132*x^20 - x^21 - 309*x^22 + 465*x^23 + x^24 + ...
%e A386659 where cubic trisections, defined by A(x)^3 = C1(x) + C2(x) + C3(x), obey
%e A386659 C3(x) = A(x^3),
%e A386659 C1(x)*C2(x) = 9*A(x^3)^3,
%e A386659 C2(x)/C1(x) = T2(x)/T1(x) = x + x^4 + 2*x^7 + 6*x^10 + 20*x^13 + 71*x^16 + 267*x^19 + 1041*x^22 + ... + A370446(n)*x^(3*n-2) + ...
%e A386659 The cubic trisections begin
%e A386659 C1(x) = 3*x^4 + 3*x^7 - 3*x^10 - 9*x^13 - 21*x^16 - 84*x^19 - 309*x^22 - 1137*x^25 - 4449*x^28 - 17868*x^31 - 73137*x^34 - 304662*x^37 - 1286388*x^40 - ...
%e A386659 C2(x) = 3*x^5 + 6*x^8 + 6*x^11 + 12*x^14 + 42*x^17 + 132*x^20 + 465*x^23 + 1791*x^26 + 7059*x^29 + 28503*x^32 + 117498*x^35 + 491757*x^38 + 2084481*x^41 + ...
%e A386659 C3(x) = x^3 + x^6 + x^15 - x^21 + x^24 - 2*x^30 + 3*x^33 - 6*x^39 + 10*x^42 - 22*x^48 + 33*x^51 + ... + a(n)*x^(3*n) + ...
%e A386659 SPECIFIC VALUES.
%e A386659 A(r) = 1 and A(r^3) = 1/4 at r = 0.591403538949431343296352603332310036448543950513103383318429...
%e A386659 A(t) = 4/5 and A(t^3) = 64/425 at t = 0.510303761967726164722767738473741580674762344121899...
%e A386659 A(t) = 3/4 and A(t^3) = 27/208 at t = 0.488075704869119285515484767956113771965332978558674...
%e A386659 A(t) = 2/3 and A(t^3) = 8/81 at t = 0.4490656139430636435247188510711544862057647445925319...
%e A386659 A(t) = 1/2 and A(t^3) = 1/20 at t = 0.3627219904933172573963798296372201737748692616169519...
%e A386659 A(t) = 1/3 and A(t^3) = 1/54 at t = 0.2629820536068200748031820994203659473004640287705972...
%e A386659 A(t) = 1/4 and A(t^3) = 1/112 at t = r^3 = 0.206848205250953970652722994332475597057157203674066...
%e A386659 A(t) = 1/5 and A(t^3) = 1/200 at t = 0.170714946526968286919515308872119424149511936479752...
%e A386659 A(1/2) = 0.7765855959847885627987696942587081429921785817514493... where A(1/8) = A(1/2)^3/(1 + 3*A(1/2)).
%e A386659 A(1/3) = 0.4482359377100401660271468423571796863698018480508060... where A(1/27) = A(1/3)^3/(1 + 3*A(1/3)).
%e A386659 A(1/4) = 0.3134295384970268001359461486249333443235800254018265... where A(1/64) = A(1/4)^3/(1 + 3*A(1/4)).
%e A386659 A(1/8) = 0.1406550988235082384593126468031209848166962450443705...
%e A386659 A(1/27) = 0.038408848749171730717291402355749106248762924579924...
%e A386659 A(1/64) = 0.015869141556098751959628853939856842544839850661716...
%o A386659 (PARI) {a(n) = my(V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
%o A386659 V[#V] = polcoef( subst(A,x, x^3) - A^3/(1 + 3*A), #V+1)/3; ); V[n+1] }
%o A386659 for(n=1,54,print1(a(n),", "))
%Y A386659 Cf. A370446, A264228.
%K A386659 sign,new
%O A386659 1,10
%A A386659 _Paul D. Hanna_, Aug 28 2025