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A028296
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E.g.f. Gudermannian(x) = 2*arctan(exp(x))-Pi/2.
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11
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1, -1, 5, -61, 1385, -50521, 2702765, -199360981, 19391512145, -2404879675441, 370371188237525, -69348874393137901, 15514534163557086905, -4087072509293123892361, 1252259641403629865468285
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The first column of the inverse to the matrix with entries C(2*i,2*j), i,j >=0. The full matrix is lower triangular with the i-th sundiagonal having entries a(i)*C(2*j,2*i) j>=i. - Nolan Wallach (nwallach(AT)ucsd.edu), Dec 26 2005
This sequence is also EulerE(2*n). - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 14 2006
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REFERENCES
| Gradshteyn and Ryzhik, Tables, 5th ed., Section 1.490, pp. 51-52.
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LINKS
| N. E. Noerlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 25.
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FORMULA
| E.g.f.: sech(x) or gd(x).
Recurrence: a(n) = -sum(i=0..n-1, a(i)*C(2*n, 2*i) ). - Ralf Stephan, Feb 24 2005
a(n) = sum_{k=1,3,5,..,2n+1} (-1)^((k-1)/2) /(2^k*k) *sum_{i=0..k} (-1)^i*(k-2*i)^(2n+1) *binomial(k,i); [From Vladimir Kruchinin, Apr 20 2011]
a(n) = 2^(4*n+1)*(zeta(-2*n,1/4)-zeta(-2*n,3/4)) - [Gerry Martens, May 27 2011]
G.f.: A(x) = 1 - x/(S(0)+x), S(k) = euler(2*k) + x*euler(2*k+2) - x*euler(2*k)*euler(2*k+4)/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
E.g.f.: E(x) = 1 - x/(S(0)+x); S(k) = (k+1)*euler(2*k) + x*euler(2*k+2) - x*(k+1)*euler(2*k)*euler(2*k+4)/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
2*arctan(exp(z))-Pi/2 = z*(1 - z^2/(G(0) + z^2)), G(k) = 2*(k+1)*(2*k+3)*euler(2*k) + z^2*euler(2*k+2) - 2*z^2*(k+1)*(2*k+3)*euler(2*k)*euler(2*k+4)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
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EXAMPLE
| gd x = x - 1/6*x^3 + 1/24*x^5 - 61/5040*x^7 + 277/72576*x^9 + ....
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MAPLE
| A028296 := proc(n) a :=0 ; for k from 1 to 2*n+1 by 2 do a := a+(-1)^((k-1)/2)/2^k/k *add( (-1)^i *(k-2*i)^(2*n+1) *binomial(k, i), i=0..k) ; end do: a ; end proc:
seq(A028296(n), n=0..10) ; # R. J. Mathar, Apr 20 2011
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MATHEMATICA
| Table[EulerE[2 n], {n, 0, 30}] (* Paul Abbott, Apr 14 2006 *)
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PROG
| (Maxima)
a(n):=sum((-1+(-1)^(k))*(-1)^((k+1)/2)/(2^(k+1)*k)*sum((-1)^i*(k-2*i)^n*binomial(k, i), i, 0, k), k, 1, n); /* with interpolated zeros, Vladimir Kruchinin, Apr 20 2011 */
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CROSSREFS
| Essentially same as A000364.
Sequence in context: A196125 A096537 A115047 * A000364 A159316 A201254
Adjacent sequences: A028293 A028294 A028295 * A028297 A028298 A028299
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KEYWORD
| sign,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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