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A340338 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^(n+1)). 7

%I

%S 1,2,4,11,35,121,444,1689,6592,26258,106313,436203,1809727,7579202,

%T 31999297,136050472,582002281,2503242025,10818689627,46959246659,

%U 204623676444,894785832949,3925297799901,17270153317728,76187650017660,336934181461844

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

%H Vaclav Kotesovec, <a href="/A340338/b340338.txt">Table of n, a(n) for n = 0..280</a>

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

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

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

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

%F a(n) ~ c * d^n / n^(3/2), where d = 4.6940696906229278555829659... and c = 0.71283897646430285525... - _Vaclav Kotesovec_, Jan 07 2021

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 11*x^3 + 35*x^4 + 121*x^5 + 444*x^6 + 1689*x^7 + 6592*x^8 + 26258*x^9 + 106313*x^10 + ...

%e where

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

%e also

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

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

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 05 2021

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Last modified June 20 06:56 EDT 2021. Contains 345157 sequences. (Running on oeis4.)