A180715 - OEIS

%I #18 Apr 30 2019 03:47:22

%S 1,2,10,84,988,14944,276288,6037088,152213344,4349539776,138913306816,

%T 4903586835328,189581185491072,7966928227397120,361586320101395968,

%U 17626603314884699136,918522989907500809216,50952388648850059964416,2997739520942089756839936

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

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

%F (1) (1+A(x))*log(1+A(x)) = (3*A(x) - x)/2.

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

%F (3) Let B(x) = 1+A(x), then: B(x) = exp( B(x)^2 * Integral 1/B(x)^3 dx ). - _Paul D. Hanna_, Dec 06 2013

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

%e E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 84*x^4/4! + 988*x^5/5! + ...

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

%e x - x^2 + x^3/3 - x^4/6 + x^5/10 - x^6/15 + x^7/21 - x^8/28 +- ...

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

%e x - x^2/2! - 3*x^3/3! - 5*x^4/4! - 7*x^5/5! - 9*x^6/6! - 11*x^7/7! - ...

%p series(exp(LambertW(-exp(-3/2)*(3+x)/2)+3/2)-1, x, 31): A:=simplify(%, symbolic): A180715:=n->n!*coeff(A, x, n): # _Vladeta Jovovic_, Sep 28 2010

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

%o for(n=1,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 06 2013

%Y Cf. A074059, A232693.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 24 2010