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 A372526 #17 May 05 2024 21:03:29
%S A372526 1,1,2,6,20,70,256,969,3762,14895,59916,244179,1006026,4183396,
%T A372526 17534888,74007851,314256048,1341575769,5754629794,24789907450,
%U A372526 107202369386,465209278326,2025212712660,8842042378050,38707067608872,169860383434800,747096961093560,3292855742992644
%N A372526 Expansion of g.f. A(x) satisfying A( x*A(x)^2 + A(x)^4 ) = A(x)^3.
%H A372526 Paul D. Hanna, <a href="/A372526/b372526.txt">Table of n, a(n) for n = 1..1030</a>
%F A372526 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A372526 (1) A(x)^3 = A( x*A(x)^2 + A(x)^4 ).
%F A372526 (2) A(x)^9 = A( x*A(x)^8 + A(x)^10 + A(x)^12 ).
%F A372526 (3) A(x)^27 = A( x*A(x)^26 + A(x)^28 + A(x)^30 + A(x)^36 ).
%F A372526 (4) A(x)^(3^n) = A( x*A(x)^(3^n-1) + Sum_{k=0..n-1} A(x)^(3^n+3^k) ) for n > 0.
%F A372526 (5) A(x) = x + Sum_{n>=0} A(x)^(3^n+1).
%F A372526 (6) A(x) = Series_Reversion(x - x*Sum_{n>=0} x^(3^n) ).
%F A372526 The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} (3^n+1) * A(r)^(3^n) and r = A(r) - Sum_{n>=0} A(r)^(3^n+1), where r = 0.214732801800375010254079407876131682823903064701286670006... and A(r) = 0.384967312289976324530970877165834568783164468488676531438...
%e A372526 G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 256*x^7 + 969*x^8 + 3762*x^9 + 14895*x^10 + 59916*x^11 + 244179*x^12 + ...
%e A372526 where A( x*A(x)^2 + A(x)^4 ) = A(x)^3.
%e A372526 RELATED SERIES.
%e A372526 (1) A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 432*x^8 + 1676*x^9 + 6633*x^10 + 26676*x^11 + 108696*x^12 + ...
%e A372526 (2) x*A(x)^2 + A(x)^4 = x^3 + 3*x^4 + 9*x^5 + 30*x^6 + 108*x^7 + 405*x^8 + 1560*x^9 + 6138*x^10 + 24570*x^11 + 99738*x^12 + ...
%e A372526 (3) Let R(x) be the series reversion of A(x), R(A(x)) = x, then
%e A372526 R(x) = x - x^2 - x^4 - x^10 - x^28 - x^82 - x^244 - x^730 + ... + -x^(3^n+1) + ...
%e A372526 SPECIFIC VALUES.
%e A372526 A(1/5) = 0.2937167157779136500722875625899113632023...
%e A372526 A(1/6) = 0.2150539986528250703029216090552606059919...
%e A372526 A(1/7) = 0.1740789503092637057579787813575613522976...
%e A372526 A(1/8) = 0.1471095742959948638409574049543396207684...
%o A372526 (PARI) /* Using series reversion of x - x*Sum_{n>=0} x^(3^n) */
%o A372526 {a(n) = my(A); A = serreverse( x - x*sum(k=0, ceil(log(n)/log(3)), x^(3^k) +x*O(x^n)) ); polcoeff(A, n)}
%o A372526 for(n=1, 35, print1(a(n), ", "))
%o A372526 (PARI) /* Using A(x)^3 = A( x*A(x)^2 + A(x)^4 ) */
%o A372526 {a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A);
%o A372526 A[#A] = polcoeff( subst(F, x, x*F^2 + F^4 ) - F^3, #A+2) ); A[n]}
%o A372526 for(n=1, 35, print1(a(n), ", "))
%Y A372526 Cf. A371713, A371716.
%K A372526 nonn
%O A372526 1,3
%A A372526 _Paul D. Hanna_, May 04 2024