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
Keywords
Examples
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! +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..250
Programs
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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)}
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
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