OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = Sum_{n>=0} x^n * ( (A(x)^sqrt(2*n) + x)^sqrt(2*n) + A(x)^(2*n)/(1 + x*A(x)^sqrt(2*n))^sqrt(2*n) )/2.
(2) A(x) = Sum_{n>=0} x^n * ( (A(x)^sqrt(2*n) + x)^sqrt(2*n) + 1/(1/A(x)^sqrt(2*n) + x)^sqrt(2*n) )/2.
EXAMPLE
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 66*x^3/3! + 1080*x^4/4! + 24210*x^5/5! + 689160*x^6/6! + 23806160*x^7/7! + 967458688*x^8/8! + 45226555164*x^9/9! + 2391009969600*x^10/10! + ...
SPECIFIC VALUES.
A(x) = sqrt(2) at x = 0.156934823890276913061148911866717130169736795253864482...
A(2/13) = 1.37929578566666389269739402377879264233420917786313229616...
A(1/7) = 1.29917227096629410743551160817342310342620062625852904026...
A(1/8) = 1.22070880501883803045142406156799735607641427440734581159...
PROG
(PARI) /* must set precision suitable for desired number of terms */
\p60
{a(n) = my(A=1+x); for(i=1, 30, A = sum(m=0, n, x^m/2 * ( ((A +x*O(x^n))^sqrt(2*m) + x)^sqrt(2*m) + A^(2*m)/(1 + x*(A +x*O(x^n))^sqrt(2*m) )^sqrt(2*m) ) )); round(n!*polcoeff(H=A, n))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 03 2023
STATUS
approved