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 A372578 #9 Jun 18 2024 10:39:26
%S A372578 1,2,10,60,406,2940,22304,174960,1407582,11550396,96299472,813433712,
%T A372578 6946442776,59872428672,520174647424,4550665293920,40052871669422,
%U A372578 354421196057404,3151211548631856,28137903707808048,252219507331523688,2268719274696321856,20472066335198022080,185268984285773695200
%N A372578 Expansion of g.f. A(x) satisfying A( x*A(x) + 2*A(x)^3 ) = A(x)^2.
%H A372578 Paul D. Hanna, <a href="/A372578/b372578.txt">Table of n, a(n) for n = 1..520</a>
%F A372578 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A372578 (1) A(x)^2 = A( x*A(x) + 2*A(x)^3 ).
%F A372578 (2) A(x)^4 = A( x*A(x)^3 + 2*A(x)^5 + 2*A(x)^6 ).
%F A372578 (3) A(x)^8 = A( x*A(x)^7 + 2*A(x)^9 + 2*A(x)^10 + 2*A(x)^12 ).
%F A372578 (4) A(x)^(2^n) = A( x*A(x)^(2^n-1) + 2*Sum_{k=0,n-1} A(x)^(2^n+2^k) ).
%F A372578 (5) A(x) = x + 2*Sum_{n>=0} A(x)^(2^n+1).
%F A372578 (6) A(x) = Series_Reversion( x - 2*x*Sum_{n>=0} x^(2^n) ).
%F A372578 The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} 2*(2^n+1) * A(r)^(2^n) and r = A(r) - 2*Sum_{n>=0} A(r)^(2^n+1), where r = 0.103594274393575546296984777950632418580281502255382627835... and A(r) = 0.191573759982214348953869719011237665707785580853712880852...
%e A372578 G.f.: A(x) = x + 2*x^2 + 10*x^3 + 60*x^4 + 406*x^5 + 2940*x^6 + 22304*x^7 + 174960*x^8 + 1407582*x^9 + 11550396*x^10 + 96299472*x^11 + 813433712*x^12 + ...
%e A372578 where A( x*A(x) + 2*A(x)^3 ) = A(x)^2.
%e A372578 RELATED SERIES.
%e A372578 A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 160*x^5 + 1152*x^6 + 8704*x^7 + 68088*x^8 + 546656*x^9 + 4478720*x^10 + 37294080*x^11 + ...
%e A372578 A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 308*x^6 + 2358*x^7 + 18612*x^8 + 150424*x^9 + 1238688*x^10 + 10355982*x^11 + 87672468*x^12 + ...
%e A372578 The series reversion R(x) of A(x), R(A(x)) = x, begins:
%e A372578 R(x) = x - 2*x^2 - 2*x^3 - 2*x^5 - 2*x^9 - 2*x^17 - 2*x^33 - 2*x^65 - 2*x^129 - 2*x^257 - 2*x^513 + ... + -2*x^(2^n+1) + ...
%e A372578 SPECIFIC VALUES.
%e A372578 A(1/10) = 0.1580645870348513671680526916072548213169829162556439...
%e A372578 A(1/11) = 0.1278454819475039498675733418966788971517121949516108...
%e A372578 A(1/12) = 0.1104694875320629136831876267359845627848091250498995...
%o A372578 (PARI) {a(n) = my(A=serreverse(x - 2*x*sum(m=0,#binary(n),x^(2^m) +x*O(x^n)))); polcoeff(A,n)}
%o A372578 for(n=1, 30, print1(a(n), ", "))
%o A372578 (PARI) {a(n) = my(A=[0,1]); for(i=1, n, A = concat(A,0); F=Ser(A); A[#A] = polcoeff( subst(F,x, x*F + 2*F^3) - F^2, #A) ); A[n+1]}
%o A372578 for(n=1, 30, print1(a(n), ", "))
%Y A372578 Cf. A371713, A356782.
%K A372578 nonn
%O A372578 1,2
%A A372578 _Paul D. Hanna_, Jun 17 2024