OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} x^n * A(x)^n * (A(x)^n + x^n)^(n+1).
(2) A(x) = Sum_{n=-oo..+oo} x^(n^2) * A(x)^(n^2) / (A(x)^(n+1) + x^(n+1))^n.
EXAMPLE
G.f.: A(x) = 1 + 3*x + 4*x^2 + 27*x^3 + 151*x^4 + 1111*x^5 + 8980*x^6 + 80071*x^7 + 775670*x^8 + 7964186*x^9 + 86714747*x^10 + ...
SPECIFIC VALUES.
At x = -1/45, the g.f. A(x) as a power series converges to
A(-1/45) = 0.935044053364283766196715891026961792781415707709299721...
At x = -1/4, A(x) diverges but may be evaluated formally to be
A(-1/4) = 0.395473062947146515629761575746517811238869867686921280...
PROG
(PARI) {a(n) = my(A=1 + x*O(x^n)); for(i=0, n, A = sum(m=-n, n, x^(m^2) * A^(m^2) / (A^(m+1) + x^(m+1) +x*O(x^n) )^m ) ); polcoeff(A, n) }
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0); A[#A] = Vec( sum(m=-#A-1, #A+1, x^m*Ser(A)^m * (Ser(A)^m + x^m)^(m+1)))[#A]); A[n+1] }
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 22 2023
STATUS
approved