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A060082
Coefficients of even-indexed Euler polynomials (falling powers without zeros).
4
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, 2073, 1, -7, 91, -1001, 7293, -31031, 62881, -38227, 1, -8, 140, -2184, 24310, -177320, 754572, -1529080, 929569, 1, -9, 204, -4284, 67626, -753610, 5497596, -23394924, 47408019
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
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).
-E(2n, -1/2)*(-4)^n/3 = A076552(n), -E(2n, 1/3)*(-9)^n/2 = A002114(n).
Cf. A060083 (rising powers), A060096-7 (Euler polynomials), A004172 (with zeros).
Columns (left edge) include A000330, A053132. Columns (right edge) include A001469.
Sequence in context: A007754 A144866 A058732 * A102225 A183262 A287030
KEYWORD
sign,easy,tabl
AUTHOR
Wolfdieter Lang, Mar 29 2001
EXTENSIONS
Edited by Ralf Stephan, Nov 05 2004
STATUS
approved