A012259 Expansion of e.g.f. exp(arctanh(tan(x))).

1, 1, 1, 5, 17, 121, 721, 6845, 58337, 698161, 7734241, 111973685, 1526099057, 25947503401, 419784870961, 8200346492525, 153563504618177, 3389281372287841, 72104198836466881, 1774459993676715365, 42270463533824671697, 1147649139272698443481

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• nonn

Examples

			 exp(arctanh(tan(x))) = 1 + x + x^2/2! + 5*x^3/3! + 17*x^4/4! + 121*x^5/5! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..425
• Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, The alternating run polynomials of permutations, arXiv:1904.11437 [math.CO], 2019.
• Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, David-Barton type identities and alternating run polynomials, Academia Sinica (Taipei, 2019).
• Mathematics Stack Exchange, Mystery regarding power series of 1/sqrt(1+x^x), Question 6939.

Crossrefs

Cf. A012077, A012085, A185411, A202038 (signed version).

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Sqrt((1+Tan(x))/(1-Tan(x))) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 06 2019

Mathematica

With[{nn=30}, CoefficientList[Series[Sqrt[(1+Tan[x])/(1-Tan[x])], {x, 0, nn}], x]*Range[0,nn]!] (* Vaclav Kotesovec, Oct 23 2013 *)

PARI

{a(n)=local(A=1); for(i=0, n, A = exp( intformal( (A^2 + subst(A^2, x, -x))/2 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2017

PARI

my(x='x+O('x^30)); Vec(serlaplace( sqrt((1+tan(x))/(1-tan(x))) )) \\ G. C. Greubel, Jun 06 2019

Sage

m = 30; T = taylor(sqrt((1+tan(x))/(1-tan(x))), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jun 06 2019

Formula

Alternative form of e.g.f: sqrt(sec(2*x) + tan(2*x)) = 1 + x + x^2/2! + 5*x^3/3! + 17*x^4/4! + ... (where sec(x)=1/cos(x)). - Peter Bala, Jan 11 2011

a(n) = 2^n*Z(n,1/2), where Z(n,x) is the n-th zigzag polynomial as defined in A147309.

Put y = x*log(x)/4. The connection between the expansion sqrt(2/(1+x^x)) = 1 - y - y^2/2! + 5*y^3/3! + 17*y^4/4! - 121*y^5/5! ... and the present sequence is explained in the answer to Mathematics Stack Exchange Question 6939. - Peter Bala, Jul 10 2011

exp(arctanh(tan(x))) = sqrt( (1 + tan(x))/(1 - tan(x) ) ) = sqrt( tan(x+pi/4) ). - David Callan, Dec 13 2011

a(n) ~ 2^(2*n+3/2) * n^n / (Pi^(n+1/2) * exp(n)). - Vaclav Kotesovec, Oct 23 2013

E.g.f. A(x) satisfies: A(x) = exp( Integral (A(x)^2 + A(-x)^2)/2 dx ). - Paul D. Hanna, Feb 04 2017

E.g.f. A(x) satisfies: A'(x) = A(x) * (A(x)^2 + A(-x)^2)/2. - Paul D. Hanna, Feb 04 2017