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Search: id:A060082
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| A060082 |
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Coefficients of even indexed Euler polynomials (falling powers without zeros). |
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+0
3
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| 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; listen)
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OFFSET
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0,5
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COMMENT
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E(2n,x) = x^(2n) + Sum[k=1..n, a(n,k)*x^(2n-2k+1) ].
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REFERENCES
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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.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
Z.-W. Sun, Introduction to Bernoulli and Euler polynomials
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FORMULA
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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.
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EXAMPLE
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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.
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PROGRAM
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(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 */
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CROSSREFS
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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: A090234 A007754 A058732 this_sequence A102225 A075248 A128325
Adjacent sequences: A060079 A060080 A060081 this_sequence A060083 A060084 A060085
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KEYWORD
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sign,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 29 2001
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EXTENSIONS
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Edited by Ralf Stephan, Nov 05 2004
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