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A215093
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E.g.f. satisfies: A(x) = x + sinh( A(x) )^2 / 2.
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1
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1, 1, 3, 19, 165, 1801, 24003, 378379, 6880485, 141757201, 3263757123, 83046239299, 2314209491685, 70093262093401, 2292753819807363, 80548997707137979, 3024937662747436965, 120925183043471954401, 5127013172890341294723, 229794790034059392232819
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OFFSET
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1,3
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COMMENTS
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Radius of convergence is r = log(sqrt(5)+2)/2 - (sqrt(5)-1)/4 = 0.41280074...,
where A(r) = log(sqrt(5)+2)/2 = arcsinh(2)/2 = 0.72181773...
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LINKS
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FORMULA
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E.g.f. satisfies:
(1) A(x) = Series_Reversion( x - sinh(x)^2/2 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x)^(2*n)/(n!*2^n).
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x)^(2*n)/(n!*2^n*x) ).
(4) A'(x) = 1/(1 - sinh(2*A(x))/2).
a(n) ~ 2^(2*n-1) * n^(n-1) / (5^(1/4) * exp(n) * (1-sqrt(5) + log(9+4*sqrt(5)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014
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EXAMPLE
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A(x) = x + x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! + ...
sinh(A(x)) = G(x) is the e.g.f. of A215094:
G(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 211*x^5/5! + 2296*x^6/6! + ...
where
G(x)^2/2 = x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! + ...
Related expansions:
A(x) = x + sinh(x)^2/2 + d/dx sinh(x)^4/(2!*2^2) + d^2/dx^2 sinh(x)^6/(3!*2^3) + d^3/dx^3 sinh(x)^8/(4!*2^4) + ...
log(A(x)/x) = sinh(x)^2/(2*x) + d/dx sinh(x)^4/(2!*2^2*x) + d^2/dx^2 sinh(x)^6/(3!*2^3*x) + d^3/dx^3 sinh(x)^8/(4!*2^4*x) + ...
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MATHEMATICA
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max = 20; Rest[ CoefficientList[ InverseSeries[ Series[x - Sinh[x]^2/2, {x, 0, max}], x], x]]*Range[max]! (* Jean-François Alcover, Aug 06 2012, from 1st formula *)
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PROG
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(PARI) {a(n)=n!*polcoeff(serreverse(x-sinh(x+x*O(x^n))^2/2), n)}
(PARI) {a(n)=local(A=x); for(i=0, n, A=x + sinh(A)^2/2); n!*polcoeff(A, 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, sinh(x+x*O(x^n))^(2*m)/(m!*2^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, sinh(x+x*O(x^n))^(2*m)/(m!*2^m*x))+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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