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A215188 E.g.f.: A(x) = x + sinh(A(x)^2). 4
1, 2, 12, 120, 1680, 30360, 672000, 17599680, 532224000, 18248660640, 699512647680, 29642193060480, 1375922515968000, 69427962935210880, 3783838462038835200, 221509040567970355200, 13862292728701236019200, 923523471334492405977600, 65257265823541297938432000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
E.g.f.: Series_Reversion(x - sinh(x^2)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x^2)^n/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*sinh(x^2)^n/n! ).
a(n) ~ n^(n-1) / (sqrt(1/s + 4*s^2*sinh(s^2)) * (s-sinh(s^2))^(n-1/2) * exp(n)), where s = 0.4863332989938055129629... is the root of the equation 2*s*cosh(s^2) = 1. - Vaclav Kotesovec, Jan 23 2014
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
where A(x - sinh(x^2)) = x and A(x) = x + sinh(A(x)^2).
Series expansions:
A(x) = x + sinh(x^2) + d/dx sinh(x^2)^2/2! + d^2/dx^2 sinh(x^2)^3/3! + d^3/dx^3 sinh(x^2)^4/4! +...
log(A(x)/x) = sinh(x^2)/x + d/dx (sinh(x^2)^2/x)/2! + d^2/dx^2 (sinh(x^2)^3/x)/3! + d^3/dx^3 (sinh(x^2)^4/x)/4! +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x - Sinh[x^2], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 23 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(serreverse(x-sinh(x^2+x^2*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, sinh(x^2+x*O(x^n))^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, sinh(x^2+x*O(x^n))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
CROSSREFS
Sequence in context: A108135 A097388 A001813 * A236357 A226759 A303557
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 05 2012
STATUS
approved

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Last modified March 2 02:30 EST 2024. Contains 370447 sequences. (Running on oeis4.)