A143134 E.g.f. satisfies: A(x) = x + sin( A(x) )^2 with A(0)=0.
1, 2, 12, 112, 1440, 23552, 467712, 10926592, 293544960, 8914583552, 301957742592, 11285975498752, 461367611228160, 20477098870833152, 980591931131953152, 50393637174029320192, 2766350676943951626240
Offset: 1
Keywords
Examples
A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +... sin(A(x)) = G(x) is the e.g.f. of A143135: G(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1261*x^5/5! +... G(x)^2 = 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +... Related expansions: A(x) = x + sin(x)^2 + d/dx sin(x)^4/2! + d^2/dx^2 sin(x)^6/3! + d^3/dx^3 sin(x)^8/4! +... log(A(x)/x) = sin(x)^2/x + d/dx (sin(x)^4/x)/2! + d^2/dx^2 (sin(x)^6/x)/3! + d^3/dx^3 (sin(x)^8/x)/4! +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..370
- V. Kotesovec, Asymptotic of implicit functions if Fww = 0, Jan 19 2014
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[x - Sin[x]^2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 18 2014 *) -
PARI
{a(n)=local(A=x);for(i=0,n,A=x + sin(A)^2);n!*polcoeff(A,n)} -
PARI
{a(n)=n!*polcoeff(serreverse(x-sin(x+x*O(x^n))^2),n)} -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, sin(x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(A, n)} -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, sin(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", "))
Formula
E.g.f. satisfies:
(1) A(x) = Series_Reversion( x - sin(x)^2 ).
(2) A(x) = x + Sum_{n>=1} (-1)^(n-1)*2^(2*n-1) * A(x)^(2*n)/(2*n)!.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x)^(2*n)/n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (sin(x)^(2*n)/x)/n! ).
E.g.f. derivative: A'(x) = 1/(1 - 2*sqrt(A(x)-x)*sqrt(1+x-A(x))); thus A'(x) = 1/(1 - sin(2*A(x))).
Let f(x) = 1/(1-sin(2*x)). Then a(n) = (f(x)*d/dx)^(n-1) f(x) evaluated at x = 0. - Peter Bala, Oct 12 2011
a(n) ~ GAMMA(1/3) * 2^(2*n-3/2) * n^(n-5/6) / (3^(1/6) * sqrt(Pi) * exp(n) * (Pi-2)^(n-1/3)). - Vaclav Kotesovec, Jan 18 2014
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