Search: id:A060082
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%I A060082
%S A060082 1,1,1,1,2,1,1,3,5,3,1,4,14,28,17,1,5,30,126,255,155,1,6,55,396,1683,3410,
%T A060082 2073,1,7,91,1001,7293,31031,62881,38227,1,8,140,2184,24310,177320,754572,
%U A060082 1529080,929569,1,9,204,4284,67626,753610,5497596,23394924,47408019
%V A060082 1,1,-1,1,-2,1,1,-3,5,-3,1,-4,14,-28,17,1,-5,30,-126,255,-155,1,-6,55,
               -396,1683,-3410,
%W A060082 2073,1,-7,91,-1001,7293,-31031,62881,-38227,1,-8,140,-2184,24310,-177320,
               754572,
%X A060082 -1529080,929569,1,-9,204,-4284,67626,-753610,5497596,-23394924,47408019
%N A060082 Coefficients of even indexed Euler polynomials (falling powers without 
               zeros).
%C A060082 E(2n,x) = x^(2n) + Sum[k=1..n, a(n,k)*x^(2n-2k+1) ].
%D A060082 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 A060082 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A060082 Z.-W. Sun, <a href="http://pweb.nju.edu.cn/zwsun/papers.htm">Introduction 
               to Bernoulli and Euler polynomials</a>
%F A060082 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 A060082 E(0,x) = 1.
%e A060082 E(2,x) = x^2 - x.
%e A060082 E(4,x) = x^4 - 2*x^3 + x.
%e A060082 E(6,x) = x^6 - 3*x^5 + 5*x^3 - 3*x.
%e A060082 E(8,x) = x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x.
%e A060082 E(10,x) = x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x.
%o A060082 (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)) /* from R. Stephan */
%Y A060082 E(2n, 1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).
%Y A060082 -E(2n, -1/2)*(-4)^n/3 = A076552(n), -E(2n, 1/3)*(-9)^n/2 = A002114(n).
%Y A060082 Cf. A060083 (rising powers), A060096-7 (Euler polynomials), A004172 (with 
               zeros).
%Y A060082 Columns (left edge) include A000330, A053132. Columns (right edge) include 
               A001469.
%Y A060082 Sequence in context: A007754 A144866 A058732 this_sequence A102225 A145236 
               A075248
%Y A060082 Adjacent sequences: A060079 A060080 A060081 this_sequence A060083 A060084 
               A060085
%K A060082 sign,easy,tabl
%O A060082 0,5
%A A060082 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 29 
               2001
%E A060082 Edited by Ralf Stephan, Nov 05 2004

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