A002112 Glaisher's H numbers.
3, 33, 903, 46113, 3784503, 455538993, 75603118503, 16546026500673, 4616979073434903, 1599868423237443153, 674014138103352845703, 339274210193051498798433, 201097637653063767131142903, 138634566390566081044811718513
Offset: 1
References
- A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 76.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..100
- J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
- Michael E. Hoffman, Derivative polynomials, Euler polynomials, and associated integer sequences, The Electronic Journal of Combinatorics 6.1 (1999).
- Index entries for sequences related to Glaisher's numbers
Programs
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Mathematica
e[0] = 1; e[n_] := e[n] = (-1)^n*(1 - Sum[(-1)^i*Binomial[2n, 2i]*3^(2n-2i)*e[i], {i, 0, n-1}]); a[n_] := 3*e[n]/2^(2n+1); Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Jan 31 2012, after Philippe Deléham *)
Formula
H(n) = 2^(2n+1)*I(n), where e.g.f. for (-1)^n*I(n) is (3/2)/(1+exp(x)+exp(-x)) (see A047788, A047789).
E.g.f.: E(x) = 3*x^2/(G(0)-x^2); G(k) = 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 03 2012
If E(x) = Sum_{k>=0} a(k+1)*x^(2k+2), then A002112(k) = a(k+1)*(2*k+2)!. - Sergei N. Gladkovskii, Jan 09 2012
From Vaclav Kotesovec, May 05 2020: (Start)
a(n) = sqrt(3) * (2*n)! * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (2*Pi)^(2*n+1).
a(n) = (-1)^(n+1) * sqrt(3) * Bernoulli(2*n) * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (4*Pi*zeta(2*n)). (End)