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A215363
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E.g.f. satisfies: A(x) = x + A(x)^2*cosh(A(x)).
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1
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1, 2, 12, 132, 2040, 40350, 974400, 27805736, 915505920, 34160797530, 1424581678080, 65660547312492, 3314551571595264, 181866769617012662, 10777121944589844480, 685937077729538151120, 46668919680893409361920, 3380042082757952844150066, 259638732115410022642483200
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. satisfies:
(1) Series_Reversion(x - x^2*cosh(x)).
(2) x + Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n) / n!.
(3) x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n-1) / n! ).
a(n) = [x^n/n!] x/(1 - x*cosh(x))^n / n for n>0.
a(n) ~ s*sqrt(r/(6*r-2*s-r*s^2+s^3)) * n^(n-1) / (exp(n) * r^n), where s = 0.4227473416936597149... is the root of the equation s*(2*cosh(s) + s*sinh(s)) = 1, and r = s - s^2*cosh(s) = 0.22782318947143997934... - Vaclav Kotesovec, Jan 13 2014
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EXAMPLE
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E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
Series expressions:
A(x) = x + cosh(x)*x^2 + d/dx cosh(x)^2*x^4/2! + d^2/dx^2 cosh(x)^3*x^6/3! + d^3/dx^3 cosh(x)^4*x^8/4! +...
log(A(x)/x) = cosh(x)*x + d/dx cosh(x)^2*x^3/2! + d^2/dx^2 cosh(x)^3*x^5/3! + d^3/dx^3 cosh(x)^4*x^7/4! +...
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MATHEMATICA
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Rest[CoefficientList[InverseSeries[Series[x - x^2*Cosh[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)
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PROG
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(PARI) {a(n)=n!*polcoeff(serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
for(n=1, 25, print1(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, cosh(x+x*O(x^n))^m*x^(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, cosh(x+x*O(x^n))^m*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(x/(1 - x*cosh(x+x*O(x^n)))^n/n, 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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