The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons 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 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A215362 E.g.f. satisfies: A(x) = x - (exp(A(x)) - 1) * log(1 - A(x)). 0
 1, 2, 18, 258, 5165, 132805, 4171552, 154816396, 6628496163, 321607254643, 17438557525290, 1045054518319394, 68589806347167547, 4893053653878222677, 376976677153445288012, 31194322614945877275400, 2759269697125674797536075, 259812433080660490261356447 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA E.g.f. satisfies: (1) A(x) = Series_Reversion(x + (exp(x)-1)*log(1-x) ). (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*(-log(1-x))^n / n!. (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*(-log(1-x))^n/x / n! ). a(n) ~ n^(n-1) * (1-s)*sqrt(1/(exp(s)*(3-2*s) - 1 - exp(s)*(1-s)^2*log(1-s))) / (exp(n) * r^(n-1/2)), where s = 0.3036870503596169812... is the root of the equation (2-s-exp(s))/(1-s) + exp(s)*log(1-s) = 0, and r = s + (exp(s)-1)*log(1-s) = 0.1752487484407525433... - Vaclav Kotesovec, Jan 13 2014 EXAMPLE E.g.f: A(x) = x + 2*x^2/2! + 18*x^3/3! + 258*x^4/4! + 5165*x^5/5! +... where A(x + (exp(x)-1)*log(1-x)) = x. The e.g.f. satisfies the series: A(x) = x - (exp(x)-1)*log(1-x) + d/dx (exp(x)-1)^2*log(1-x)^2/2! - d^2/dx^2 (exp(x)-1)^3*log(1-x)^3/3! + d^3/dx^3 (exp(x)-1)^4*log(1-x)^4/4! +... Logarithmic series: log(A(x)/x) = -(exp(x)-1)*log(1-x)/x + d/dx (exp(x)-1)^2*log(1-x)^2/x/2! - d^2/dx^2 (exp(x)-1)^3*log(1-x)^3/x/3! + d^3/dx^3 (exp(x)-1)^4*log(1-x)^4/x/4! +... MATHEMATICA Rest[CoefficientList[InverseSeries[Series[x + (Exp[x]-1)*Log[1-x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *) PROG (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x + (exp(x+x*O(x^n))-1)*log(1-x+x*O(x^n))))); n!*polcoeff(A, n)} for(n=1, 21, print1(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)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*(-log(1-x+x*O(x^n)))^m)/m!)+x*O(x^n)); 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)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*(-log(1-x+x*O(x^n)))^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)} CROSSREFS Sequence in context: A213643 A143138 A151362 * A099880 A141009 A143154 Adjacent sequences:  A215359 A215360 A215361 * A215363 A215364 A215365 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 08 2012 STATUS approved

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

Last modified December 1 01:37 EST 2021. Contains 349426 sequences. (Running on oeis4.)