OFFSET
0,5
COMMENTS
E(2n,x) = x^(2n) + Sum_{k=1..n} a(n,k)*x^(2n-2k+1).
REFERENCES
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.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
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.
EXAMPLE
E(0,x) = 1.
E(2,x) = x^2 - x.
E(4,x) = x^4 - 2*x^3 + x.
E(6,x) = x^6 - 3*x^5 + 5*x^3 - 3*x.
E(8,x) = x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x.
E(10,x) = x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x.
MATHEMATICA
Table[ CoefficientList[ EulerE[2*n, x], x] // Reverse // DeleteCases[#, 0]&, {n, 0, 9}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
PROG
(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
CROSSREFS
E(2n, 1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).
KEYWORD
AUTHOR
Wolfdieter Lang, Mar 29 2001
EXTENSIONS
Edited by Ralf Stephan, Nov 05 2004
STATUS
approved