OFFSET
0,3
COMMENTS
Compare to: C(x) = 1 + x*C(x) * (C(x) + 2*x*C'(x)) / (C(x) + x*C'(x)) holds when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) A(x) = 1 + x*A(x) * (A(x) + 3*x*A'(x)) / (A(x) + x*A'(x)).
(2) A(x) = 1/(1 - x*A(x)/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - 5*x*A(x)/(1 - ...)))))), a continued fraction.
(3) A(x) = Series_Reversion( x - x^2*F(x) ) where F(x) = Sum_{n>=0} (2*n)!/(n!*2^n)*x^n (g.f. of the odd double factorials A001147).
a(n) ~ 2^(n - 1/2) * n^(n-1) / exp(n - 1/2). - Vaclav Kotesovec, Aug 11 2021
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 71*x^4 + 469*x^5 + 3711*x^6 + 35181*x^7 + 398791*x^8 + 5352149*x^9 + 83650687*x^10 + ...
RELATED SERIES.
A'(x)/A(x) = 1 + 5*x + 31*x^2 + 225*x^3 + 1891*x^4 + 18473*x^5 + 210939*x^6 + 2815137*x^7 + 43551715*x^8 + 770297385*x^9 + ...
A(x) + x*A'(x) = 1 + 2*x + 9*x^2 + 52*x^3 + 355*x^4 + 2814*x^5 + 25977*x^6 + 281448*x^7 + 3589119*x^8 + 53521490*x^9 + ...
PROG
(PARI) /* Differential equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A*(A + 3*x*A')/(x*A +x^2*O(x^n))'); polcoeff(G=A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, for(k=0, n, CF = 1/(1 - (n-k+1)*x*A*CF ) ); A=1/(1 - x*A*CF) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 11 2018
STATUS
approved