A143154 E.g.f.: A(x) = x + log(1 - A(x))^2.

1, 2, 18, 262, 5320, 138728, 4419156, 166319424, 7221397848, 355312006392, 19537581248592, 1187337791554176, 79025863405440432, 5716937001401316000, 446654003380859659488, 37480492611898380241248

Offset

1,2

Comments

Radius of convergence is r = (-1 + 6*A(r) - A(r)^2)/4 = 0.172815973872...

where A(r) = 1 - exp((A(r)-1)/2) = 0.2965325775...

Links

Table of n, a(n) for n=1..16.

Formula

E.g.f.: A(x) = Series_Reversion( x - log(1 - x)^2 ).

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) log(1-x)^(2*n)/n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (log(1-x)^(2*n)/x)/n! ).

E.g.f. derivative: A'(x) = (1 - A(x))/(1 - A(x) + 2*log(1 - A(x))).

a(n) = ((n-1)!*sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, (-1)^(n+j-1)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(2*(j-l))!*stirling1(n-l+j-1,2*(j-l)))/(n-l+j-1)!)))), n>0. - Vladimir Kruchinin, Feb 07 2012

a(n) ~ c*sqrt(4/(1+c)-2-2*c) * n^(n-1) / (exp(n) * (1-c*(2+c))^n), where c = LambertW(1/2) = 0.35173371124919... (see A202356). - Vaclav Kotesovec, Jan 24 2014

Example

A(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! + ...
-log(1 - A(x)) = G(x) = the g.f. of A143155:
G(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! + ...
G(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! + ...
Related expansions:
A(x) = x + log(1-x)^2 + d/dx log(1-x)^4/2! + d^2/dx^2 log(1-x)^6/3! + d^3/dx^3 log(1-x)^8/4! + ...
log(A(x)/x) = log(1-x)^2/x + d/dx (log(1-x)^4/x)/2! + d^2/dx^2 (log(1-x)^6/x)/3! + d^3/dx^3 (log(1-x)^8/x)/4! + ...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x-Log[1-x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 24 2014 *)

Prog

(PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + log(1-A)^2); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(serreverse(x-log(1-x+x*O(x^n))^2), 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, log(1-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, log(1-x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(Maxima) a(n):=((n-1)!*sum(binomial(n+k-1, n-1)*sum((-1)^(n+j-1)*binomial(k, j)*sum((binomial(j, l)*(2*(j-l))!*stirling1(n-l+j-1, 2*(j-l)))/(n-l+j-1)!, l, 0, j), j, 0, k), k, 0, n-1)); /* Vladimir Kruchinin, Feb 07 2012 */

Crossrefs

Cf. A143155, A143138, A202356.

Sequence in context: A350461 A099880 A141009 * A032037 A351276 A138275

Adjacent sequences: A143151 A143152 A143153 * A143155 A143156 A143157

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2008

Status

approved