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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 (list; table; graph; refs; listen; history; text; internal format)
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].

Z.-W. Sun, Introduction to Bernoulli and Euler polynomials

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-Fran├žois 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

Adjacent sequences:  A060079 A060080 A060081 * A060083 A060084 A060085

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Mar 29 2001

EXTENSIONS

Edited by Ralf Stephan, Nov 05 2004

STATUS

approved

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Last modified October 23 09:28 EDT 2019. Contains 328345 sequences. (Running on oeis4.)