A290579 E.g.f. satisfies: A(x) = (1 + sin(x*A(x))) / cos(x*A(x)).

1, 1, 3, 17, 145, 1661, 23947, 416765, 8504865, 199200985, 5268047411, 155282897065, 5048546868209, 179483438440021, 6927036355244475, 288439692143197397, 12889578930617144641, 615304922810783744561, 31249797537098034966499, 1682478707717022082757441, 95721167135792642044679505, 5738213812249917154871122861, 361520344958259229318337826923, 23881324878748364254507881173677

Offset

0,3

Comments

Equals row sums of irregular triangle A290580.

A diagonal in square array A185414.

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

Cf. A000111, A290580, A185414.

Sequence in context: A368236 A162650 A015735 * A140983 A241805 A368233

Adjacent sequences: A290576 A290577 A290578 * A290580 A290581 A290582

Keyword

nonn

Author

Paul D. Hanna, Aug 06 2017

Status

approved