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
Keywords
Examples
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! + ...
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[x-Log[1-x]^2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 24 2014 *) -
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 */
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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), ", "))
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
Comments