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 A012259 E.g.f. exp(arctanh(tan(x))) = 1 + x + x^2/2! + 5*x^3/3! + 17*x^4/4! + 121*x^5/5! + ... 5
 1, 1, 1, 5, 17, 121, 721, 6845, 58337, 698161, 7734241, 111973685, 1526099057, 25947503401, 419784870961, 8200346492525, 153563504618177, 3389281372287841, 72104198836466881, 1774459993676715365, 42270463533824671697, 1147649139272698443481 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS G. C. Greubel, Table of n, a(n) for n = 0..425 Shi-Mei Ma, Jun Ma, Yeong-Nan Yeh, The alternating run polynomials of permutations, arXiv:1904.11437 [math.CO], 2019. Shi-Mei Ma, Jun Ma, Yeong-Nan Yeh, David-Barton type identities and alternating run polynomials, Academia Sinica (Taipei, 2019). StackExchange, 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 Math StackExchange 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 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:=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 A012263 Adjacent sequences:  A012256 A012257 A012258 * A012260 A012261 A012262 KEYWORD nonn AUTHOR Patrick Demichel (patrick.demichel(AT)hp.com) STATUS approved

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Last modified January 16 18:17 EST 2022. Contains 350376 sequences. (Running on oeis4.)