cp's OEIS Frontend

A370440 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3), with A(0)=0, A'(0)=1.

%I A370440 #22 Mar 14 2024 08:05:55
%S A370440 1,1,1,1,2,6,15,30,55,113,274,683,1596,3547,7990,18968,46530,113663,
%T A370440 273392,656421,1598270,3951520,9827565,24411649,60599823,150978177,
%U A370440 378293690,951828992,2398983638,6051008950,15284145261,38690832455,98154905623,249390491237,634296702273
%N A370440 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3), with A(0)=0, A'(0)=1.
%C A370440 Compare the g.f. to the following identities:
%C A370440 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
%C A370440 (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C A370440 where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H A370440 Paul D. Hanna, <a href="/A370440/b370440.txt">Table of n, a(n) for n = 1..969</a>
%F A370440 G.f. A(x) = Sum_{n>=1} a(n) * x^n satisfies the following formulas.
%F A370440 (1) A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3).
%F A370440 (2) B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where A(B(x)) = x.
%F A370440 a(n) ~ c * d^n / n^(3/2), where d = 2.653503750287... and c = 0.193303... - _Vaclav Kotesovec_, Mar 14 2024
%e A370440 G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 15*x^7 + 30*x^8 + 55*x^9 + 113*x^10 + 274*x^11 + 683*x^12 + 1596*x^13 + 3547*x^14 + 7990*x^15 + ...
%e A370440 where A(x)^3 = A( x^3 + 3*x^2*A(x)^2 ).
%e A370440 RELATED SERIES.
%e A370440 A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 18*x^7 + 47*x^8 + 106*x^9 + 216*x^10 + 450*x^11 + 1040*x^12 + ...
%e A370440 A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 109*x^9 + 264*x^10 + 585*x^11 + 1270*x^12 + ...
%e A370440 Let B(x) denote the series reversion of A(x), A(B(x)) = x,
%e A370440 B(x) = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 - 419*x^22 + 1041*x^24 - 1648*x^25 + 4168*x^27 - 6652*x^28 + 17047*x^30 + ...
%e A370440 then B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where
%e A370440 B(x)^2 = x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 3*x^6 - 3*x^8 + 4*x^9 - 8*x^11 + 11*x^12 - 23*x^14 + 34*x^15 + ...
%e A370440 B(x)^3 = x^3 - 3*x^4 + 6*x^5 - 10*x^6 + 12*x^7 - 9*x^8 + x^9 + 9*x^10 - 12*x^11 - x^12 + 24*x^13 - 33*x^14 + 69*x^16 - 102*x^17 + ...
%e A370440 Further, the trisections of B(x) = C1(x) + C2(x) + C3(x) are
%e A370440 C1(x) = x^4/C3(x) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 - ...
%e A370440 C2(x) = -x^2, and
%e A370440 C3(x) = x^3 + x^6 + 2*x^9 + 6*x^12 + 20*x^15 + 71*x^18 + 267*x^21 + 1041*x^24 + 4168*x^27 + 17047*x^30 + 70902*x^33 + ... + A370446(n)*x^(3*n) + ...
%e A370440 Compare these series to the series trisections involved in series reversion of A264228.
%e A370440 SPECIFIC VALUES.
%e A370440 A(1/3) = 0.5339969110985873619406256103732700685272...
%e A370440 A(1/4) = 0.3373018860609501862067597141160425025580...
%e A370440 A(1/5) = 0.2509433336474255853462277222741392614966...
%e A370440 A(1/6) = 0.2003115176013404351183299069966738623357...
%e A370440 A(1/8) = 0.1429156905534693639298206599148805278651...
%e A370440 A(1/3)^3 = A(1/27 + 3*A(1/3)^2/9) = A(0.132087937391...) = 0.152270661558...
%e A370440 A(1/4)^3 = A(1/64 + 3*A(1/4)^2/16) = A(0.036957355438...) = 0.038375699859...
%e A370440 A(1/5)^3 = A(1/125 + 3*A(1/5)^2/25) = A(0.015556706804...) = 0.250943333647...
%o A370440 (PARI) {a(n) = my(A=[1],G); for(i=1,n, G = x*Ser(A); A = Vec((subst(G,x, x^3 + 3*x^2*G^2) + x^4*O(x^#A))^(1/3)); );A[n+1]}
%o A370440 for(n=0,40, print1(a(n),", "))
%Y A370440 Cf. A370441, A370446, A264228, A356781.
%K A370440 nonn
%O A370440 1,5
%A A370440 _Paul D. Hanna_, Mar 09 2024