A009752 Expansion of e.g.f. tan(x)*x (even powers only).
0, 2, 8, 96, 2176, 79360, 4245504, 313155584, 30460116992, 3777576173568, 581777702256640, 108932957168730112, 24370173276164456448, 6419958484945407574016, 1967044844910430876860416, 693575525634287935244206080, 278846808228005417477465964544, 126799861926498005417315327279104
Offset: 0
Keywords
Examples
2*x/(1+e^(2*x)) = 0 + x - 2/2!*x^2 + 8/4!*x^4 - 96/6!*x^6 + 2176/8!*x^8 ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..240
- Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232 [math.CO], 2022. See p. 12.
Programs
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Maple
a := n -> 4^n*n*`if`(n=0,0,abs(euler(2*n-1, 0))): # Peter Luschny, Jun 09 2016
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Mathematica
nn = 30; t = Range[0, nn]! CoefficientList[Series[x*Tan[x], {x, 0, nn}], x]; Take[t, {1, nn + 1, 2}] (* T. D. Noe, Sep 20 2012 *) Table[(-1)^n 4 n PolyLog[1 - 2 n, -I], {n, 0, 19}] (* Peter Luschny, Aug 17 2021 *) -
PARI
my(x='x+O('x^50)); v=Vec(serlaplace(x*tan(x))); concat([0], vector(#v\2,n,v[2*n-1])) \\ G. C. Greubel, Feb 12 2018
Formula
a(n) = n 4^n |E_{2n-1}(1/2)+E_{2n-1}(1)| for n > 0; E_{n}(x) Euler polynomials. - Peter Luschny, Nov 25 2010
a(n) = (2*n)! * [x^(2*n)] tan(x)*x.
a(n) = 2*(2*n)!*Pi^(-2*n)*(4^n-1)*Li{2*n}(1) for n > 0. - Peter Luschny, Jun 29 2012
E.g.f.: sqrt(x)*tan(sqrt(x))= sum(n>=0, a(n)*x^n/(2*n)! ) = x/T(0) where T(k)= 1 - 4*k^2 + x*(1 - 4*k^2)/T(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 19 2012
E.g.f.: -1 - x^(1/2)- Q(0),where Q(k) = 4*k -1 - x/( 1 - x/ (4*k+1 + x/( 1 + x/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2013
From Peter Luschny, Jun 09 2016: (Start)
a(n) = (4^n-16^n)*Sum_{k=0..2*n} (-1)^(n-k)*Stirling2(2*n, k)*k!/(k+1).
2*a(n)/4^n = A110501(n) for n>=1.
a(n) / 2^n = A117513(n) for n>=1. (End)
a(n) ~ (4*(4^(2*n)-2^(2*n)))*Pi*(n/(Pi*e))^(2*n+1/2)*exp(1/2+1/(24*n)-1/(2880*n^3) +1/(40320*n^5)-...). - Peter Luschny, Jan 16 2017
a(n) = (-1)^n*4*n*PolyLog(1 - 2*n, -i). - Peter Luschny, Aug 17 2021
a(n) = 2*A024255(n). - Alois P. Heinz, Aug 17 2021
Extensions
Extended and signs tested by Olivier Gérard, Mar 15 1997