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

0,4

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.

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

Example

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

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 *)

Prog

(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
(Magma) m:=30; R<x>:=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
(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

Crossrefs

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

Sequence in context: A324411 A012174 A202038 * A256459 A113936 A365295

Adjacent sequences: A012256 A012257 A012258 * A012260 A012261 A012262

Keyword

nonn

Author

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

Status

approved