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 A385909 #8 Aug 04 2025 04:08:43
%S A385909 1,1,3,9,31,122,493,2086,9106,40764,186206,865068,4076020,19437711,
%T A385909 93655043,455293416,2230636436,11003483165,54607084364,272453502850,
%U A385909 1365876088389,6876896373019,34757806185051,176291771193079,897001780346928,4577362669389502,23420275560794225,120123996076924029
%N A385909 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^(3*n+1).
%C A385909 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.
%H A385909 Paul D. Hanna, <a href="/A385909/b385909.txt">Table of n, a(n) for n = 0..300</a>
%F A385909 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A385909 (1) 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).
%F A385909 (2) 0 = Sum_{n=-oo..+oo} x^(3*n*(2*n-1)) / (1 - A(x)*x^(2*n))^(3*n-1).
%F A385909 a(n) ~ c * d^n / n^(3/2), where d = 5.437310827623... and c = 0.230314472... - _Vaclav Kotesovec_, Aug 04 2025
%e A385909 G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 31*x^4 + 122*x^5 + 493*x^6 + 2086*x^7 + 9106*x^8 + 40764*x^9 + 186206*x^10 + 865068*x^11 + 4076020*x^12 + ...
%o A385909 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A385909 A[#A] = polcoeff( sum(n=-#A, #A, x^n*(x^(2*n) - Ser(A))^(3*n+1) ), #A-1)); A[n+1]}
%o A385909 for(n=0, 30, print1(a(n), ", "))
%Y A385909 Cf. A355866.
%K A385909 nonn
%O A385909 0,3
%A A385909 _Paul D. Hanna_, Jul 29 2025