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 A002112 Glaisher's H numbers. (Formerly M3135 N1272) 4
 3, 33, 903, 46113, 3784503, 455538993, 75603118503, 16546026500673, 4616979073434903, 1599868423237443153, 674014138103352845703, 339274210193051498798433, 201097637653063767131142903, 138634566390566081044811718513 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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). 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). H(n) = 3* A000436(n)/2^(2n+1)= 3*A002114(n). - Philippe Deléham, Jan 17 2004 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=0,1,..., 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) 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 *) CROSSREFS Sequence in context: A124432 A234715 A126466 * A055549 A086894 A255930 Adjacent sequences:  A002109 A002110 A002111 * A002113 A002114 A002115 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified June 24 21:09 EDT 2021. Contains 345425 sequences. (Running on oeis4.)