A228608 E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^4.
1, 2, 12, 128, 1968, 39488, 977088, 28742912, 979744512, 37968868352, 1648597834752, 79272057049088, 4181485522464768, 240067201819885568, 14902137637759008768, 994529776192394166272, 71009035425186633940992, 5401058272888913168433152, 435991257271370763778916352
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 1968*x^4/4! + 39488*x^5/5! +... Related expansions. A(x)^2 = 1 + 4*x + 32*x^2/2! + 400*x^3/3! + 6848*x^4/4! + 149056*x^5/5! +... A(x)^4 = 1 + 8*x + 96*x^2/2! + 1568*x^3/3! + 32640*x^4/4! + 828032*x^5/5! +... The logarithm of e.g.f. A(x) begins: log(A(x)) = 2*x + 8*x^2/2! + 72*x^3/3! + 992*x^4/4! + 18336*x^5/5! +... and equals Integral A(x) + A(x)^3 dx, where A(x)^3 = 1 + 6*x + 60*x^2/2! + 864*x^3/3! + 16368*x^4/4! + 385344*x^5/5! +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..350
Programs
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Mathematica
CoefficientList[Exp[InverseSeries[Series[1-Exp[-x]-ArcTan[Tanh[x/2]], {x, 0, 20}], x]],x]*Range[0, 20]! (* Vaclav Kotesovec, Dec 20 2013 *) -
PARI
/* Explicit formula: */ {a(n)=local(A,X=x+x^2*O(x^n));A=exp(serreverse(1-exp(-X) - atan(tanh(X/2))));n!*polcoeff(A,n)} for(n=0,20,print1(a(n),", ")) -
PARI
/* By definition: A'(x) = A(x)^2 + A(x)^4: */ {a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A^2+A^4+x*O(x^n))); n!*polcoeff(A, n)} for(n=0,20,print1(a(n),", ")) -
PARI
/* From: A(x) = exp( Integral A(x) + A(x)^3 dx ): */ {a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(A+A^3)+x*O(x^n))); n!*polcoeff(A, n)} for(n=0,20,print1(a(n),", "))
Formula
E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) + A(x)^3 dx ) with A(0)=1.
(2) A(x) = (1 + B(x))/(1 - B(x)) where B(x) = tan(1-x - 1/A(x)).
(3) log(A(x)) = Series_Reversion( 1-exp(-x) - atan(tanh(x/2)) ).
(4) A( 1-exp(-x) - atan(tanh(x/2)) ) = exp(x).
a(n) ~ n! / (GAMMA(1/3) * 3^(1/3) * n^(2/3) * (1-Pi/4)^(n+1/3)). - Vaclav Kotesovec, Jan 26 2014