A359923 - OEIS

%I #9 Jan 21 2023 18:44:32

%S 1,1,6,15,69,376,1741,8860,46044,245074,1336538,7337135,40736876,

%T 228625148,1293530435,7372491383,42275811853,243742895280,

%U 1412310750812,8219298313118,48023377286364,281592177442072,1656522460985914,9773791391488278,57824226906859849

%N a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (3 + x^n*A(x)^n)^n.

%H Paul D. Hanna, <a href="/A359923/b359923.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (3 + x^n*A(x)^n)^n.

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

%e G.f.: A(x) = 1 + x + 6*x^2 + 15*x^3 + 69*x^4 + 376*x^5 + 1741*x^6 + 8860*x^7 + 46044*x^8 + 245074*x^9 + 1336538*x^10 + ...

%e where

%e x = ... + x^6*A(x)^9/(1 + 3*x^3*A(x)^3)^3 - x^2*A(x)^4/(1 + 3*x^2*A(x)^2)^2 + A(x)/(1 + 3*x*A(x)) - 1 + x*(3 + x*A(x)) - x^2*(3 + x^2*A(x)^2)^2 + x^3*(3 + x^3*A(x)^3)^3 + ... + (-1)^(n-1) * x^n * (3 + x^n*A(x)^n)^n + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);

%o A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^(m-1) * x^m * (3 + (x*Ser(A))^m)^m ), #A-1)); A[n+1]}

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

%Y Cf. A359672, A359922.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 18 2023