A215368 E.g.f.: Series_Reversion( x*cos(x) - x*sin(x) ).

1, 2, 15, 176, 2905, 61536, 1592703, 48706048, 1718376561, 68702272000, 3069734553743, 151592011714560, 8198710703202825, 481965222651551744, 30598546651134134655, 2086474763912627879936, 152083996930329322871521, 11800530001358902191587328, 971113004536128839898536079

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Formula

E.g.f. satisfies: A(x) = x / (cos(A(x)) - sin(A(x))).

a(n) = [x^n/n!] 1/(cos(x)-sin(x))^n / n.

a(n) = n*A201923(n-1).

a(n) ~ sqrt(-1 + 4/(3 + sin(2*s))) * n^(n-1) / (r^n * exp(n)), where s = 0.4026281741881116098199325239112307245635064777960... is the root of the equation s*cos(2*s) + sin(2*s) = 1 and r = s*(cos(s) - sin(s)) = 0.21266685344074710045360679397024815598865409988038310855608986167... - Vaclav Kotesovec, Oct 04 2020

Example

E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 176*x^4/4! + 2905*x^5/5! +...
where A(x*cos(x) - x*sin(x)) = x and A(x) = x/(cos(A(x)) - sin(A(x))).
Related expansions:
cos(A(x)) = 1 - x^2/2! - 6*x^3/3! - 71*x^4/4! - 1160*x^5/5! - 24481*x^6/6! - 631904*x^7/7! - 19288079*x^8/8! -...
sin(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 164*x^4/4! + 2696*x^5/5! + 57006*x^6/6! + 1473632*x^7/7! + 45026344*x^8/8! +...

Prog

(PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff(serreverse(x*cos(X)-x*sin(X)), n)}
(PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff(x/(cos(X)-sin(X))^n/n, n)}
for(n=1, 31, print1(a(n), ", "))

Crossrefs

Cf. A201923, A215638.

Sequence in context: A360483 A222920 A036080 * A200795 A121427 A364340

Adjacent sequences: A215365 A215366 A215367 * A215369 A215370 A215371

Keyword

nonn

Author

Paul D. Hanna, Aug 08 2012

Status

approved