OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
E.g.f.: A(x) = (1/x) * Series_Reversion( x*cos(x)/(1 + sin(x)) ).
E.g.f. satisfies: A(x) = Sum_{n>=0} A000111(n) * x^n * A(x)^n / n!.
E.g.f.: A(x) = exp( sum_{n>=1} A185414(n,n)*x^n/n! ).
a(n) = A185414(n,n+1) for n>=1.
a(n) = [x^n/n!] ( (1 + sin(x)) / cos(x) )^(n+1) / (n+1) for n>=0.
a(n) ~ n^(n-1) / (exp(n) * r^(n+1/2)), where r = 0.3263879708167851846572540343... is the root of the equation r = 1 - sin(sqrt(r*(2-r))). - Vaclav Kotesovec, Aug 20 2017
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 145*x^4/4! + 1661*x^5/5! + 23947*x^6/6! + 416765*x^7/7! + 8504865*x^8/8! +...
such that A(x) = (1 + sin(x*A(x))) / cos(x*A(x)).
RELATED SERIES.
(1 + sin(x)) / cos(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 16*x^5/5! + 61*x^6/6! + 272*x^7/7! + 1385*x^8/8! +...+ A000111(n)*x^n/n! +...
where A000111 is the sequence of zigzag numbers.
Given e.g.f. A(x), then the logarithm of A(x) begins:
log(A(x)) = x + 2*x^2/2! + 10*x^3/3! + 80*x^4/4! + 880*x^5/5! + 12336*x^6/6! + 210320*x^7/7! + 4225024*x^8/8! +...+ A185414(n,n)*x^n/n! +...
PROG
(PARI) {a(n) = my(A=1); for(i=0, n, A = (1 + sin(x*A)) / cos(x*A +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(X=x +O(x^(n+2))); A = (1/x) * serreverse( x*cos(X)/(1 + sin(X)) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 06 2017
STATUS
approved