The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A143136 E.g.f. satisfies: A(x) = x + sinh( A(x) )^2. 5
 1, 2, 12, 128, 1920, 36992, 870912, 24232448, 777999360, 28309164032, 1151292628992, 51750540443648, 2547747292446720, 136336755956252672, 7879446478581399552, 489119124160488931328, 32456290094449950720000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Radius of convergence is r = log(sqrt(2)+1)/2 - (sqrt(2)-1)/2 = 0.2335800..., where A(r) = log(1+sqrt(2))/2 = arcsinh(1)/2 = 0.44068679... LINKS Alois P. Heinz, Table of n, a(n) for n = 1..150 FORMULA E.g.f.: A(x) = 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) (sinh(x)^(2*n)/x)/n! ). E.g.f. derivative: A'(x) = 1/(1 - sinh(2*A(x))). a(n) ~ 2^(n-5/4) * n^(n-1) / (exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014 EXAMPLE A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ... sinh(A(x)) = G(x) is the e.g.f. of A143137: G(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2101*x^5/5! + ... Related expansions: A(x) = x + sinh(x)^2 + d/dx sinh(x)^4/2! + d^2/dx^2 sinh(x)^6/3! + d^3/dx^3 sinh(x)^8/4! + ... log(A(x)/x) = sinh(x)^2/x + d/dx (sinh(x)^4/x)/2! + d^2/dx^2 (sinh(x)^6/x)/3! + d^3/dx^3 (sinh(x)^8/x)/4! +... MATHEMATICA Rest[CoefficientList[InverseSeries[Series[x - Sinh[x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *) PROG (PARI) {a(n)=n!*polcoeff(serreverse(x-sinh(x+x*O(x^n))^2), n)} (PARI) {a(n)=local(A=x); for(i=0, n, A=x + sinh(A)^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!)); 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)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) CROSSREFS Cf. A143134, A143137, A213643. Sequence in context: A201470 A349268 A003712 * A214224 A214431 A227461 Adjacent sequences: A143133 A143134 A143135 * A143137 A143138 A143139 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 27 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)