A060082 - OEIS

%I #26 Jul 05 2018 02:02:18

%S 1,1,-1,1,-2,1,1,-3,5,-3,1,-4,14,-28,17,1,-5,30,-126,255,-155,1,-6,55,

%T -396,1683,-3410,2073,1,-7,91,-1001,7293,-31031,62881,-38227,1,-8,140,

%U -2184,24310,-177320,754572,-1529080,929569,1,-9,204,-4284,67626,-753610,5497596,-23394924,47408019

%N Coefficients of even-indexed Euler polynomials (falling powers without zeros).

%C E(2n,x) = x^(2n) + Sum_{k=1..n} a(n,k)*x^(2n-2k+1).

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.

%H Vincenzo Librandi, <a href="/A060082/b060082.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Z.-W. Sun, <a href="http://pweb.nju.edu.cn/zwsun/papers.htm">Introduction to Bernoulli and Euler polynomials</a>

%F E(n, x) = 2/(n+1) * [B(n+1, x) - 2^(n+1)*B(n+1, x/2) ], with B(n, x) the Bernoulli polynomials.

%e E(0,x) = 1.

%e E(2,x) = x^2 - x.

%e E(4,x) = x^4 - 2*x^3 + x.

%e E(6,x) = x^6 - 3*x^5 + 5*x^3 - 3*x.

%e E(8,x) = x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x.

%e E(10,x) = x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x.

%t Table[ CoefficientList[ EulerE[2*n, x], x] // Reverse // DeleteCases[#, 0]&, {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Jun 21 2013 *)

%o (PARI) {B(n,v='x)=sum(i=0,n,binomial(n,i)*bernfrac(i)*v^(n-i))} E(n,v='x)=2/(n+1)*(B(n+1,v)-2^(n+1)*B(n+1,v/2)) \\ _Ralf Stephan_, Nov 05 2004

%Y E(2n, 1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).

%Y -E(2n, -1/2)*(-4)^n/3 = A076552(n), -E(2n, 1/3)*(-9)^n/2 = A002114(n).

%Y Cf. A060083 (rising powers), A060096-7 (Euler polynomials), A004172 (with zeros).

%Y Columns (left edge) include A000330, A053132. Columns (right edge) include A001469.

%K sign,easy,tabl

%O 0,5

%A _Wolfdieter Lang_, Mar 29 2001

%E Edited by _Ralf Stephan_, Nov 05 2004