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A002112 Glaisher's H numbers.
(Formerly M3135 N1272)
4

%I M3135 N1272

%S 3,33,903,46113,3784503,455538993,75603118503,16546026500673,

%T 4616979073434903,1599868423237443153,674014138103352845703,

%U 339274210193051498798433,201097637653063767131142903,138634566390566081044811718513

%N Glaisher's H numbers.

%D 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.

%D J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A002112/b002112.txt">Table of n, a(n) for n = 1..100</a>

%H J. W. L. Glaisher, <a href="http://plms.oxfordjournals.org/content/s1-31/1/216.extract">On a set of coefficients analogous to the Eulerian numbers</a>, Proc. London Math. Soc., 31 (1899), 216-235.

%H Michael E. Hoffman, <a href="http://www.emis.ams.org/journals/EJC/Volume_6/PDF/v6i1r21.pdf">Derivative polynomials, Euler polynomials, and associated integer sequences</a>, The Electronic Journal of Combinatorics [electronic only] 6.1 (1999)

%H <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>

%F 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).

%F H(n) = 3* A000436(n)/2^(2n+1)= 3*A002114(n). - _Philippe Deléham_, Jan 17 2004

%F 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

%F 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

%t 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_ *)

%K nonn,nice,easy

%O 1,1

%A _N. J. A. Sloane_.

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Last modified March 23 14:17 EDT 2019. Contains 321431 sequences. (Running on oeis4.)