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A232693 E.g.f.: A(x) = Series_Reversion( x - 3*Sum_{n>=2} (-x)^n/(n*(n-1)) ). 1
1, 3, 24, 321, 6012, 144783, 4261716, 148255893, 5951045484, 270729816075, 13765317295716, 773577708886377, 47613664279309788, 3185462518517039463, 230164306993740992436, 17862339610423895715837, 1481845528640328826524876, 130864004355639214251335907 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
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.: (4+x)/(3*LambertW(-(4+x)*exp(-4/3)/3))-1. - Vaclav Kotesovec, Dec 07 2013
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! -...
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
Sequence in context: A343445 A264561 A003236 * A319939 A082166 A354259
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2013
STATUS
approved

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Last modified May 27 03:56 EDT 2024. Contains 372847 sequences. (Running on oeis4.)