A143154 -id:A143154 - cp's OEIS frontend

A143155 E.g.f.: A(x) = -log(1 - x - A(x)^2).

1, 3, 26, 376, 7614, 198248, 6309092, 237291388, 10297903920, 506495785632, 27842563031304, 1691646018671376, 112569103111005072, 8142200129607522288, 636046143210331062048, 53366672768969064921024

Offset: 1

Paul D. Hanna, Jul 27 2008

nonn

			A(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! +...
x + A(x)^2 = 1 - exp(-A(x)) = G(x) = g.f. of A143154:
G(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...

Cf. A143139, A143154, A202356.

Table[Sum[(n+k-1)!*Sum[(-1)^(n+j-1)/(k-j)!*Sum[(StirlingS2[n-2*l+j-1,j-l])/(l!*(n-2*l+j-1)!),{l,0,Min[j,(n+j-1)/2]}],{j,0,k}],{k,0,n-1}],{n,1,20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Dec 28 2013 *)

a(n):=(sum((n+k-1)!*sum((-1)^(n+j-1)/(k-j)!*sum((stirling2(n-2*l+j-1,j-l))/(l!*(n-2*l+j-1)!),l,0,min(j,(n+j-1)/2)),j,0,k),k,0,n-1)); /* Vladimir Kruchinin, Feb 03 2012 */

{a(n)=local(A=x+O(x^n));for(i=0,n,A=-log(1-x-A^2));n!*polcoeff(A,n)}

{a(n)=n!*polcoeff(-log(1-serreverse(x-log(1-x+x*O(x^n))^2)),n)}

a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^(n+j-1)/(k-j)!)*Sum_{L=0..min(j, (n+j-1)/2)} Stirling2(n-2*L+j-1, j-l)/(L!*(n-2*L+j-1)!), n > 0. - Vladimir Kruchinin, Feb 03 2012

a(n) ~ n^(n-1) / (sqrt(2*(1+c)) * exp(n) * (1-2*c-c^2)^(n-1/2)), where c = LambertW(1/2). - Vaclav Kotesovec, Dec 28 2013

A214759 E.g.f. satisfies: A(x) = x - log(1-A(x))*log(1+A(x)).

Original entry on oeis.org

1, 2, 12, 130, 1980, 38828, 931308, 26409696, 864340080, 32064955512, 1329619343712, 60942831972240, 3059502805907184, 166958986644198240, 9840268295795334528, 622945100439880885920, 42156597800781887551296, 3036974549700781049394240, 232047627350715685524211392

Offset: 1

Paul D. Hanna, Jul 31 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 130*x^4/4! + 1980*x^5/5! +...
where A(x - log(1-x)*log(1+x)) = x and A(x) = x - log(1-A(x))*log(1+A(x)).
Related expansions:
-log(1-x)*log(1+x) = 2*x^2/2! + 10*x^4/4! + 188*x^6/6! + 7656*x^8/8! + 541152*x^10/10! + 58800960*x^12/12! +...
-log(1-A(x)) = x + 3*x^2/2! + 20*x^3/3! + 220*x^4/4! + 3374*x^5/5! + 66428*x^6/6! +...
log(1+A(x)) = x + x^2/2! + 8*x^3/3! + 88*x^4/4! + 1354*x^5/5! + 26708*x^6/6! +...
Other series:
A(x) = x - log(1-x)*log(1+x) + d/dx log(1-x)^2*log(1+x)^2/2! - d^2/dx^2 log(1-x)^3*log(1+x)^3/3! + d^3/dx^3 log(1-x)^4*log(1+x)^4/4! +...
log(A(x)/x) = -log(1-x)*log(1+x)/x + d/dx log(1-x)^2*log(1+x)^2/x/2! - d^2/dx^2 log(1-x)^3*log(1+x)^3/x/3! + d^3/dx^3 log(1-x)^4*log(1+x)^4/x/4! +...

Cf. A143154.

Rest[CoefficientList[InverseSeries[Series[x + Log[1-x]*Log[1+x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

{a(n)=n!*polcoeff(serreverse(x+log(1-x+x*O(x^n))*log(1+x+x*O(x^n))), n)}
for(n=1, 25, print1(a(n), ", "))

{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)))^m*log(1+x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

{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)))^m*log(1+x+x*O(x^n))^m/x/m!))); n!*polcoeff(A, n)}

E.g.f.: Series_Reversion(x + log(1-x)*log(1+x)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x))^n*log(1+x)^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x))^n*log(1+x)^n/x / n! ).
a(n) ~ n^(n-1) * sqrt(r*(1-s^2) / (2+s + log(1-s)/(1+s) + log(1+s)/(1-s))) / (exp(n) * r^n), where s = 0.4236731503498759... is the root of the equation log(1+s)/(1-s) - log(1-s)/(1+s) = 1, and r = s + log(1-s)*log(1+s) = 0.2290093920590019... - Vaclav Kotesovec, Jan 13 2014

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A143155 E.g.f.: A(x) = -log(1 - x - A(x)^2).

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A214759 E.g.f. satisfies: A(x) = x - log(1-A(x))*log(1+A(x)).

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Mathematica

PARI

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