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
Keywords
Examples
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! +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..300
Programs
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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),", "))
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