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A232693 E.g.f.: A(x) = Series_Reversion( x - 3*Sum_{n>=2} (-x)^n/(n*(n-1)) ). 1

%I #13 Dec 08 2013 06:23:39

%S 1,3,24,321,6012,144783,4261716,148255893,5951045484,270729816075,

%T 13765317295716,773577708886377,47613664279309788,3185462518517039463,

%U 230164306993740992436,17862339610423895715837,1481845528640328826524876,130864004355639214251335907

%N E.g.f.: A(x) = Series_Reversion( x - 3*Sum_{n>=2} (-x)^n/(n*(n-1)) ).

%C Generally, for (1+A(x))*log(1+A(x))=((p+1)*A(x)-x)/p, E.g.f.: (1+p+x)/(p*LambertW(-((exp(-1-1/p)*(1+p+x))/p)))-1, a(n) ~ n^(n-1) / (sqrt(p) * exp(n-1/(2*p)) * (p*exp(1/p)-p-1)^(n-1/2)). - _Vaclav Kotesovec_, Dec 07 2013

%F E.g.f. A(x) satisfies:

%F (1) (1+A(x))*log(1+A(x)) = (4*A(x) - x)/3.

%F (2) log(1+A(x)) = Series_Reversion[(4-3*x)*exp(x) - 4].

%F (3) Let B(x) = 1+A(x), then: B(x) = exp( B(x)^3 * Integral 1/B(x)^4 dx ).

%F E.g.f.: (4+x)/(3*LambertW(-(4+x)*exp(-4/3)/3))-1. - _Vaclav Kotesovec_, Dec 07 2013

%F a(n) ~ n^(n-1) / (sqrt(3) * exp(n-1/6) * (3*exp(1/3)-4)^(n-1/2)). - _Vaclav Kotesovec_, Dec 07 2013

%e E.g.f.: A(x) = x + 3*x^2/2! + 24*x^3/3! + 321*x^4/4! + 6012*x^5/5! +...

%e Series reversion of the e.g.f. A(x) begins:

%e x - 3*x^2/2 + 3*x^3/6 - 3*x^4/12 + 3*x^5/20 - 3*x^6/30 + 3*x^7/42 - 3*x^8/56 +-...

%e Series reversion of log(1+A(x)) begins:

%e x - 2*x^2/2! - 5*x^3/3! - 8*x^4/4! - 11*x^5/5! - 14*x^6/6! - 17*x^7/7! -...

%o (PARI) {a(n)=if(n<1, 0, n!*polcoeff(serreverse(x-3*sum(k=2, n, (-x)^k/(k*(k-1)))+x*O(x^n)), n))}

%o for(n=1,25,print1(a(n),", "))

%o (PARI) {a(n)=if(n<1,0,n!*polcoeff(exp(serreverse((4-3*x)*exp(x+x*O(x^n))-4))-1,n))}

%o for(n=1,25,print1(a(n),", "))

%o (PARI) {a(n)=local(B=1+x);for(i=1,n,B=exp(B^3*intformal(1/B^4+x*O(x^n))));n!*polcoeff(B-1,n)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A180715, A074059.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 06 2013

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