A009752 - OEIS

%I #64 Aug 04 2024 01:26:20

%S 0,2,8,96,2176,79360,4245504,313155584,30460116992,3777576173568,

%T 581777702256640,108932957168730112,24370173276164456448,

%U 6419958484945407574016,1967044844910430876860416,693575525634287935244206080,278846808228005417477465964544,126799861926498005417315327279104

%N Expansion of e.g.f. tan(x)*x (even powers only).

%H G. C. Greubel, <a href="/A009752/b009752.txt">Table of n, a(n) for n = 0..240</a>

%H Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2212.07232">Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers</a>, arXiv:2212.07232 [math.CO], 2022. See p. 12.

%F 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

%F a(n) = (2*n)! * [x^(2*n)] tan(x)*x.

%F a(n) = 2*(2*n)!*Pi^(-2*n)*(4^n-1)*Li{2*n}(1) for n > 0. - _Peter Luschny_, Jun 29 2012

%F 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

%F 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

%F From _Peter Luschny_, Jun 09 2016: (Start)

%F a(n) = (4^n-16^n)*Sum_{k=0..2*n} (-1)^(n-k)*Stirling2(2*n, k)*k!/(k+1).

%F 2*a(n)/4^n = A110501(n) for n>=1.

%F a(n) / 2^n = A117513(n) for n>=1. (End)

%F 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

%F a(n) = (-1)^n*4*n*PolyLog(1 - 2*n, -i). - _Peter Luschny_, Aug 17 2021

%F a(n) = 2*A024255(n). - _Alois P. Heinz_, Aug 17 2021

%e 2*x/(1+e^(2*x)) = 0 + x - 2/2!*x^2 + 8/4!*x^4 - 96/6!*x^6 + 2176/8!*x^8 ...

%p a := n -> 4^n*n*`if`(n=0,0,abs(euler(2*n-1, 0))): # _Peter Luschny_, Jun 09 2016

%t 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 *)

%t Table[(-1)^n 4 n PolyLog[1 - 2 n, -I], {n, 0, 19}] (* _Peter Luschny_, Aug 17 2021 *)

%o (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

%Y A diagonal of A232933.

%Y Cf. A009725, A024255, A065619, A099028, A110501, A117513.

%K nonn

%O 0,2

%A _R. H. Hardin_

%E Extended and signs tested by _Olivier Gérard_, Mar 15 1997