A340355 - OEIS

%I #13 Feb 20 2021 15:02:39

%S 1,1,4,15,73,373,2036,11518,67108,399596,2421477,14883232,92561649,

%T 581401130,3683031612,23502839520,150944260610,974905750378,

%U 6328238976958,41261578774953,270119042203681,1774773646080126,11699419572947070,77355905980770122,512885661648043804

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^(2*n)*A(x)^n / (1 - x^(n+1)*A(x)^(n+2)).

%C The g.f. A(x) of this sequence is motivated by the following identity:

%C Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;

%C here, p = x^2*A(x), q = x*A(x)^2, and r = x*A(x).

%H Vaclav Kotesovec, <a href="/A340355/b340355.txt">Table of n, a(n) for n = 0..240</a>

%F G.f. A(x) satisfies the following relations.

%F (1) A(x) = Sum_{n>=0} x^(2*n) * A(x)^n / (1 - x^(n+1)*A(x)^(n+2)).

%F (2) A(x) = Sum_{n>=0} x^n * A(x)^(2*n) / (1 - x^(n+2)*A(x)^(n+1)).

%F (3) A(x) = Sum_{n>=0} x^(n^2+3*n) * A(x)^(n^2+3*n) * (1 - x^(2*n+3)*A(x)^(2*n+3)) / ((1 - x^(n+1)*A(x)^(n+2))*(1 - x^(n+2)*A(x)^(n+1))).

%F a(n) ~ c * d^n / n^(3/2), where d = 7.060918158410189777854181567407... and c = 0.2611318997628883837033125... - _Vaclav Kotesovec_, Jan 07 2021

%e G.f.: A(x) =  1 + x + 4*x^2 + 15*x^3 + 73*x^4 + 373*x^5 + 2036*x^6 + 11518*x^7 + 67108*x^8 + 399596*x^9 + 2421477*x^10 + ...

%e where

%e A(x) = 1/(1 - x*A(x)^2) + x^2*A(x)/(1 - x^2*A(x)^3) + x^4*A(x)^2/(1 - x^3*A(x)^4) + x^6*A(x)^3/(1 - x^4*A(x)^5) + x^8*A(x)^4/(1 - x^5*A(x)^6) + ...

%e also

%e A(x) = 1/(1 - x^2*A(x)) + x*A(x)^2/(1 - x^3*A(x)^2) + x^2*A(x)^4/(1 - x^4*A(x)^3) + x^3*A(x)^6/(1 - x^5*A(x)^4) + x^4*A(x)^8/(1 - x^6*A(x)^5) + ...

%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^(2*m)*A^m / (1 - x^(m+1)*A^(m+2) +x*O(x^n)) )); polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m*A^(2*m) / (1 - x^(m+2)*A^(m+1) +x*O(x^n)) )); ; polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 05 2021