A341380 - OEIS

%I #13 Mar 03 2021 02:41:16

%S 1,2,2,6,19,58,198,680,2410,8695,31870,118377,444315,1683400,6428086,

%T 24715541,95603500,371784813,1452687192,5700329627,22454015652,

%U 88755923251,351944894971,1399612973849,5580765692117,22306991852511,89365286885821

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

%H Vaclav Kotesovec, <a href="/A341380/b341380.txt">Table of n, a(n) for n = 0..250</a>

%F G.f. A(x) satisfies:

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

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

%F (3) A(x) = Sum_{n>=0} x^(2*n*(n+1)) * (1 - x^(4*n+2)) / ((1 - x^(2*n+1)*A(x))*(1 - x^(2*n+1)/A(x)).

%F a(n) ~ c * d^n / n^(3/2), where d = 4.24149290765489... and c = 0.598891272666... - _Vaclav Kotesovec_, Mar 03 2021

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 19*x^4 + 58*x^5 + 198*x^6 + 680*x^7 + 2410*x^8 + 8695*x^9 + 31870*x^10 + 118377*x^11 + 444315*x^12 + ...

%e where

%e A(x) = 1/(1 - x/A(x)) + x*A(x)/(1 - x^3/A(x)) + x^2*A(x)^2/(1 - x^5/A(x)) + x^3*A(x)^3/(1 - x^7/A(x)) + x^4*A(x)^4/(1 - x^9/A(x)) + ...

%e also

%e A(x) = 1/(1 - x*A(x)) + (x/A(x))/(1 - x^3*A(x)) + (x/A(x))^2/(1 - x^5*A(x)) + (x/A(x))^3/(1 - x^7*A(x)) + (x/A(x))^4/(1 - x^9*A(x)) + ...

%o (PARI) {a(n) = my(A=1); for(i=1,n,

%o A = sum(m=0,n, x^m*A^m/(1 - x^(2*m+1)/A +x*O(x^n)) ););

%o polcoeff(A,n)}

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

%o (PARI) {a(n) = my(A=1); for(i=1,n,

%o A = sum(m=0,n, x^m/A^m/(1 - x^(2*m+1)*A +x*O(x^n)) ););

%o polcoeff(A,n)}

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

%o (PARI) {a(n) = my(A=1); for(i=1,n,

%o A = sum(m=0,sqrtint(n), x^(2*m*(m+1)) * (1 - x^(4*m+2)) / ((1 - x^(2*m+1)*A)*(1 - x^(2*m+1)/A) +x*O(x^n)) ););

%o polcoeff(A,n)}

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 25 2021