A215368 - OEIS

%I #12 Oct 04 2020 11:21:12

%S 1,2,15,176,2905,61536,1592703,48706048,1718376561,68702272000,

%T 3069734553743,151592011714560,8198710703202825,481965222651551744,

%U 30598546651134134655,2086474763912627879936,152083996930329322871521,11800530001358902191587328,971113004536128839898536079

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

%H Vaclav Kotesovec, <a href="/A215368/b215368.txt">Table of n, a(n) for n = 1..300</a>

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

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

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

%F 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

%e E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 176*x^4/4! + 2905*x^5/5! +...

%e where A(x*cos(x) - x*sin(x)) = x and A(x) = x/(cos(A(x)) - sin(A(x))).

%e Related expansions:

%e 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! -...

%e 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! +...

%o (PARI) {a(n)=local(X=x+x^2*O(x^n));n!*polcoeff(serreverse(x*cos(X)-x*sin(X)),n)}

%o (PARI) {a(n)=local(X=x+x^2*O(x^n));n!*polcoeff(x/(cos(X)-sin(X))^n/n,n)}

%o for(n=1,31,print1(a(n),", "))

%Y Cf. A201923, A215638.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 08 2012