A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx.

1, 24, 32256, 285272064, 8967114326016, 735868743566229504, 130778914961055994085376, 44390350317502907443360825344, 26290393222157669992962395876622336, 25377887922329300948014930852183837507584, 37855568618678541873143615775486954119570128896

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..112

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 = 4 p. 8.

Formula

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

(1) A(x) = Series_Reversion( Integral 1/(1+x^4) dx ).

(2) A(x) = sqrt( tan( 2 * Integral A(x) dx ) ).

Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ) is the e.g.f. of A258900, then e.g.f. A(x) of this sequence satisfies:

(3) A(x) = S(x)/C(x),

(4) A(x) = Integral 1/C(x)^4 dx,

(5) A(x)^2 = S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).

a(n) ~ 2^(6*n + 14/3) * (4*n)! * n^(1/3) / (3^(1/3) * Gamma(1/3) * Pi^(4*n + 4/3)). - Vaclav Kotesovec, Jun 18 2015

Example

E.g.f.: A(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! + 735868743566229504*x^21/21! +...
where Series_Reversion(A(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! - 8310698364852576*x^21/21! +...+ A258900(n)*x^(4*n+1)/(4*n+1)! +...
C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! - 8310698364852576*x^20/20! +...+ A258900(n)*x^(4*n)/(4*n)! +...
such that C(x)^4 + S(x)^4 = 1.

Mathematica

nmax=20; Table[CoefficientList[InverseSeries[Series[Integrate[1/(1+x^4), x], {x, 0, 4*nmax+1}], x], x][[4*n-2]] * (4*n-3)!, {n, 1, nmax+1}] (* Vaclav Kotesovec, Jun 18 2015 *)

Prog

(PARI) /* E.g.f. Series_Reversion( Integral 1/(1+x^4) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^4 + O(x^(4*n+2))) ) ); (4*n+1)!*polcoeff(A, 4*n+1)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* E.g.f. A(x) = Integral 1 + A(x)^4 dx.: */
{a(n) = local(A=x); for(i=1, n+1, A = intformal( 1 + A^4 + O(x^(4*n+2)) )); (4*n+1)!*polcoeff(A, 4*n+1)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A000182, A000831, A258900, A258880, A258925, A258927, A259112, A259113, A258970.

Sequence in context: A153303 A272095 A188952 * A062322 A100733 A158664

Adjacent sequences: A258898 A258899 A258900 * A258902 A258903 A258904

Keyword

nonn

Author

Paul D. Hanna, Jun 14 2015

Status

approved