A258880 E.g.f. satisfies: A(x) = Integral 1 + A(x)^3 dx.

1, 6, 540, 184680, 157600080, 270419925600, 816984611467200, 3971317527112003200, 29097143353353192480000, 305823675529741700675520000, 4435486895868663971869188480000, 86036822683997062842122964537600000, 2175352015640142857526698650779456000000

Offset: 0

Author

Paul D. Hanna, Jun 13 2015

Keywords

• nonn

Comments

Note: Sum_{n>=0} (-1)^n*x^(3*n+1)/(3*n+1) = log( (1+x)/(1-x^3)^(1/3) )/2 + Pi*sqrt(3)/18 - atan( (1-2*x)*sqrt(3)/3 )*sqrt(3)/3.

Examples

			E.g.f.: A(x) = x + 6*x^4/4! + 540*x^7/7! + 184680*x^10/10! + 157600080*x^13/13! + 270419925600*x^16/16! +...
where Series_Reversion(A(x)) =  x - x^4/4 + x^7/7 - x^10/10 + x^13/13 - x^16/16 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..150
• Guo-Niu Han, Jing-Yi Liu, Divisibility properties of the tangent numbers and its generalizations, arXiv:1707.08882 [math.CO], 2017. See Table for k = 3 p. 8.

Crossrefs

Cf. A000182, A000831, A258878, A258901, A258925, A258927, A259112, A259113, A258969.

Programs

Mathematica

terms = 13;
A[_] = 0;
Do[A[x_] = Integrate[1 + A[x]^3, x] + O[x]^k // Normal, {k, 1, 3 terms}];
DeleteCases[CoefficientList[A[x], x] Range[0, 3 terms - 2]!, 0] (* Jean-François Alcover, Jul 25 2018 *)

PARI

{a(n) = local(A=x); A = serreverse( sum(m=0,n, (-1)^m * x^(3*m+1)/(3*m+1) ) +O(x^(3*n+2)) ); (3*n+1)!*polcoeff(A,3*n+1)}
for(n=0,20,print1(a(n),", "))

PARI

/* E.g.f. A(x) = Integral 1 + A(x)^3 dx.: */
{a(n) = local(A=x); for(i=1,n+1, A = intformal( 1 + A^3 + O(x^(3*n+2)) )); (3*n+1)!*polcoeff(A,3*n+1)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f.: Series_Reversion( Integral 1/(1+x^3) dx ).

E.g.f.: Series_Reversion( Sum_{n>=0} (-1)^n * x^(3*n+1)/(3*n+1) ).

a(n) ~ 3^(15*n/2 + 17/4) * n^(3*n+1) / (exp(3*n) * (2*Pi)^(3*n+3/2)). - Vaclav Kotesovec, Jun 15 2015