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

1, 3, 24, 321, 6012, 144783, 4261716, 148255893, 5951045484, 270729816075, 13765317295716, 773577708886377, 47613664279309788, 3185462518517039463, 230164306993740992436, 17862339610423895715837, 1481845528640328826524876, 130864004355639214251335907

Offset

1,2

Comments

Generally, for (1+A(x))*log(1+A(x))=((p+1)*A(x)-x)/p, p > 2/3, E.g.f.: 1 - (1+p+x)/(p*LambertW(-1, -exp(-1-1/p)*(1+p+x)/p)), 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

Links

Table of n, a(n) for n=1..18.

Formula

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

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

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

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

E.g.f.: 1 - (4+x)/(3*LambertW(-1, -(4+x)*exp(-4/3)/3)). - Vaclav Kotesovec, Dec 07 2013, corrected Jan 20 2026

a(n) ~ n^(n-1) / (sqrt(3) * exp(n-1/6) * (3*exp(1/3)-4)^(n-1/2)). - Vaclav Kotesovec, Dec 07 2013

Example

E.g.f.: A(x) = x + 3*x^2/2! + 24*x^3/3! + 321*x^4/4! + 6012*x^5/5! +...
Series reversion of the e.g.f. A(x) begins:
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 +-...
Series reversion of log(1+A(x)) begins:
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! -...

Mathematica

Rest[Assuming[{x > 0}, CoefficientList[Series[1 - (4 + x)/(3*LambertW[-1, -(4 + x)*E^(-4/3)/3]), {x, 0, 20}], x] * Range[0, 20]!]] (* Vaclav Kotesovec, Jan 20 2026 *)

Prog

(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))}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n)=if(n<1, 0, n!*polcoeff(exp(serreverse((4-3*x)*exp(x+x*O(x^n))-4))-1, n))}
for(n=1, 25, print1(a(n), ", "))
(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)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A180715, A074059.

Sequence in context: A343445 A264561 A003236 * A319939 A375869 A082166

Adjacent sequences: A232690 A232691 A232692 * A232694 A232695 A232696

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2013

Status

approved