A200560 E.g.f.: arcsin(x) o x/(1-x) o sin(x), a composition of functions involving sin(x) and its inverse.

1, 2, 6, 28, 180, 1502, 15456, 189208, 2683920, 43263962, 780807456, 15593180788, 341340941760, 8126644655222, 209050212857856, 5777935570510768, 170755837008595200, 5373097909706399282, 179351443518333574656, 6329687401322560131148, 235491796312126982538240

Offset

1,2

Comments

Given e.g.f. A(x), then A(Pi/6) = Pi/2, where Pi/6 is the radius of convergence.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..250

Formula

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

The n-th iteration of e.g.f. A(x) equals: arcsin(x) o x/(1-n*x) o sin(x) = arcsin( sin(x)/(1-n*sin(x)) ).

a(n) ~ 2^n * 3^(n-1/4) * n^(n-1) / (Pi^(n-1/2) * exp(n)). - Vaclav Kotesovec, Apr 05 2016

Example

E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! +...
where the initial iterations of e.g.f. A(x) begin:
A(A(x)) = arcsin( sin(x)/(1-2*sin(x)) ); more explicitly,
A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 200*x^4/4! + 2160*x^5/5! +...
A(A(A(x))) = arcsin( sin(x)/(1-3*sin(x)) ); more explicitly,
A(A(A(x))) = x + 6*x^2/2! + 54*x^3/3! + 660*x^4/4! + 10260*x^5/5! +...
A(A(A(A(x)))) = arcsin( sin(x)/(1-4*sin(x)) ); more explicitly,
A(A(A(A(x)))) = x + 8*x^2/2! + 96*x^3/3! + 1552*x^4/4! + 31680*x^5/5! +...

Prog

(PARI) {a(n)=n!*polcoeff(subst(asin(x+x*O(x^n)), x, subst(x/(1-x), x, sin(x+x*O(x^n)))), n)}

Crossrefs

Sequence in context: A276911 A104018 A100526 * A303344 A355205 A196555

Adjacent sequences: A200557 A200558 A200559 * A200561 A200562 A200563

Keyword

nonn

Author

Paul D. Hanna, Nov 29 2011

Status

approved