This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370446 #21 Mar 13 2024 11:01:17
%S A370446 1,1,2,6,20,71,267,1041,4168,17047,70902,298967,1275141,5491504,
%T A370446 23846271,104295430,459023543,2031459236,9034769573,40358643042,
%U A370446 180998556943,814645632727,3678542796070,16659932961647,75657738747396,344446195875766,1571786529601990,7187790264787872
%N A370446 Expansion of g.f. A(x) satisfying A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2.
%H A370446 Paul D. Hanna, <a href="/A370446/b370446.txt">Table of n, a(n) for n = 1..800</a>
%F A370446 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370446 (1) A(x)^3 + 2*x^2 + x^4/A(x)^3 = A(x^3) - x^2 + x^4/A(x^3).
%F A370446 (2) F( A(x^3) - x^2 + x^4/A(x^3) ) = x, where F(x) = F( x^3 + 3*x^2*F(x)^2 )^(1/3) is the g.f. of A370440.
%F A370446 (3) G( -x^2/(A(-x^3) - x^2 + x^4/A(-x^3)) ) = x, where G(x) = G( x^3/(1 - 3*x) )^(1/3) is the g.f. of A264228.
%F A370446 a(n) ~ c * d^n / n^(3/2), where d = 4.8344630246454026903035642546835542141482126303313357979263... and c = 0.0713578385738499677445741870058758452888939567284935382... - _Vaclav Kotesovec_, Mar 13 2024
%F A370446 The radius of convergence r = 0.20684820525095397... = 1/d (where d is given above), and A(r) = 0.3497581458819115559285308998459940399916633464611700768... satisfy A(r) = r^(2/3) and A(r^3) = (5 - sqrt(21))/2 * r^2. - _Paul D. Hanna_, Mar 13 2024
%e A370446 G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 267*x^7 + 1041*x^8 + 4168*x^9 + 17047*x^10 + 70902*x^11 + 298967*x^12 + 1275141*x^13 + 5491504*x^14 + 23846271*x^15 + ...
%e A370446 RELATED SERIES.
%e A370446 We can illustrate the formulas with the following related expansions.
%e A370446 (1) A(x)^3 + 2*x^2 + x^4/A(x)^3 = 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 + ...
%e A370446 which equals A(x^3) - x^2 + x^4/A(x^3), as can be seen from
%e A370446 x^4/A(x^3) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 + ...
%e A370446 A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 435*x^8 + 1715*x^9 + ...
%e A370446 x^4/A(x)^3 = x - 3*x^2 - 4*x^4 - 9*x^5 - 30*x^6 - 115*x^7 - 435*x^8 - 1713*x^9 + ...
%e A370446 (2) Let F(x) be the g.f. of A370440, which begins
%e A370446 F(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 + ...
%e A370446 where F(x)^3 = F( x^3 + 3*x^2*F(x)^2 ),
%e A370446 then the series reversion of F(x) begins
%e A370446 A(x^3) - x^2 + x^4/A(x^3) = 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 + ...
%e A370446 (3) Let G(x) be the g.f. of A264228, which begins
%e A370446 G(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 +...
%e A370446 where G(x)^3 = G( x^3/(1 - 3*x) ),
%e A370446 then the series reversion of G(x) begins
%e A370446 -x^2/(A(-x^3) - x^2 + x^4/A(-x^3)) = x^2/(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 +...).
%e A370446 SPECIFIC VALUES.
%e A370446 A(1/4.834464) = 0.349644497578571280258023712232522068793519739...
%e A370446 A(1/5) = 0.29940801195429552263938628184744484915469836164855...
%e A370446 A(1/6) = 0.21539123666426270273178791857213676628593723946879...
%e A370446 A(1/7) = 0.17414937372444126736977770687571455113383911571251...
%e A370446 A(1/8) = 0.14713126344900776621336355426627444003268957268553...
%e A370446 A(1/5^3) = 0.00806504925055020701973761348380106375185943151538...
%e A370446 A(1/6^3) = 0.00465126435780731657600811126033650347236250831668...
%e A370446 A(1/7^3) = 0.00292400175440295890949208907819991271975334925594...
%e A370446 which may be used to verify that the formula
%e A370446 A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2
%e A370446 holds for these specific values.
%o A370446 (PARI) {a(n) = my(A=x); for(m=1,n, A=truncate(A) +x^4*O(x^m); A = ( x^4/(x^4/subst(A,x,x^3) + subst(A,x,x^3) - A^3 - 3*x^2) +x^4*O(x^n))^(1/3) );polcoeff(A,n)}
%o A370446 for(n=1,30,print1(a(n),", "))
%Y A370446 Cf. A370440, A264228.
%K A370446 nonn
%O A370446 1,3
%A A370446 _Paul D. Hanna_, Mar 09 2024