Search: id:A002439
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%I A002439 M5138 N2228
%S A002439 1,23,1681,257543,67637281,27138236663,15442193173681,
%T A002439 11828536957233383,11735529528739490881,14639678925928297567703,
%U A002439 22427641105413135505628881,41393949926819051111431239623
%N A002439 Glaisher's T numbers.
%D A002439 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 A002439 J. W. L. Glaisher, Messenger of Math., 28 (1898), 36-79, see esp. p. 
               76.
%D A002439 J. W. L. Glaisher, On the Bernoullian function, Q. J. Pure Appl. Math., 
               29 (1898), 1-168.
%D A002439 J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian 
               numbers, Proc. London Math. Soc., 31 (1899), 216-235.
%D A002439 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002439 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002439 T. D. Noe, <a href="b002439.txt">Table of n, a(n) for n=0..100</a>
%H A002439 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>
%H A002439 <a href="Sindx_Ge.html#Glaisher">Index entries for sequences related 
               to Glaisher's numbers</a>
%H A002439 G. E. Andrews, J. Jimenez-Urroz, K. Ono, <a href="http://www.math.wisc.edu/
               ~ono/reprints/055.pdf">q-series identities and values of certain 
               L-functions</a>, Duke Math J., Volume 108, No.3 (2001), 395-419. 
               [From Peter Bala (pbala(AT)talktalk.net), Mar 24 2009]
%F A002439 Q_{2n+1)(sqrt(3))/sqrt(3), where the polynomials Q_n() are defined in 
               A104035. - njas, Nov 06 2009
%F A002439 E.g.f.: sin(2*x)/(2*cos(3*x)) = Sum a(n)*x^(2*n-1)/(2*n-1)!.
%F A002439 With offset 1 instead of 0: a(1)=1, a(n)=(-4)^(n-1) - Sum_{k=1..n} (-9)^k*C(2*n-1, 
               2*k)*a(n-k).
%F A002439 a(n) = (-1)^(n+1)*(1/12)*E_{2n+1}*6^(2*n+1), where E_m are the Euler 
               numbers. - R. W. Gosper Aug 08, 2001
%F A002439 Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 24 2009: (Start)
%F A002439 Basic hypergeometric generating function: exp(-t)*Sum {n = 0..inf} Product 
               {k = 1..n} (1-exp(-24*k*t)) = 1 + 23*t + 1681*t^2/2! + .... For other 
               sequences with generating functions of a similar type see A000364, 
               A000464, A002105, A079144, A158690.
%F A002439 a(n) = (1/2)*(-1)^(n+1)*L(-2*n-1), where L(s) is a Dirichlet L-function 
               for a Dirichlet character with modulus 12: L(s) = 1 - 1/5^s - 1/7^s 
               + 1/11^s + - - + .... See the Andrew's link. (End)
%p A002439 With offset 1 instead of 0: A002439:=proc(n) option remember; if n=1 
               then 1 else (-4)^(n-1) - add( (-9)^k*binomial(2*n-1, 2*k)*A002439(n-k),
               k=1..n); fi; end;
%o A002439 (PARI) a(n)=if(n<2,n>0,(-4)^(n-1)-sum(k=1,n,(-9)^k*C(2*n-1,2*k)*a(n-k)))
%Y A002439 Cf. A000364, A000464, A002105, A079144, A158690. [From Peter Bala (pbala(AT)talktalk.net), 
               Mar 24 2009]
%Y A002439 Bisections: A156175, A156176. Twice this sequence gives A000191.
%Y A002439 Sequence in context: A049003 A003281 A034243 this_sequence A132395 A064016 
               A138735
%Y A002439 Adjacent sequences: A002436 A002437 A002438 this_sequence A002440 A002441 
               A002442
%K A002439 nonn,easy,nice,new
%O A002439 0,2
%A A002439 N. J. A. Sloane (njas(AT)research.att.com).
%E A002439 More terms from Michael Somos

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