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 A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx. 10

%I

%S 1,24,32256,285272064,8967114326016,735868743566229504,

%T 130778914961055994085376,44390350317502907443360825344,

%U 26290393222157669992962395876622336,25377887922329300948014930852183837507584,37855568618678541873143615775486954119570128896

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

%H Vaclav Kotesovec, <a href="/A258901/b258901.txt">Table of n, a(n) for n = 0..112</a>

%H Guo-Niu Han, Jing-Yi Liu, <a href="https://arxiv.org/abs/1707.08882">Divisibility properties of the tangent numbers and its generalizations</a>, arXiv:1707.08882 [math.CO], 2017. See Table for k = 4 p. 8.

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

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

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

%F 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:

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

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

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

%F 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

%e 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! +...

%e where Series_Reversion(A(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 +...

%e Also, A(x) = S(x)/C(x) where

%e 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)! +...

%e 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)! +...

%e such that C(x)^4 + S(x)^4 = 1.

%t 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 *)

%o (PARI) /* E.g.f. Series_Reversion( Integral 1/(1+x^4) dx ): */

%o {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)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) /* E.g.f. A(x) = Integral 1 + A(x)^4 dx.: */

%o {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)}

%o for(n=0,20,print1(a(n),", "))

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 14 2015

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Last modified October 16 12:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)