Search: id:A002111
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%I A002111 M4007 N1660
%S A002111 1,5,49,809,20317,722813,34607305,2145998417,167317266613,16020403322021,
%T A002111 1848020950359841,252778977216700025,40453941942593304589,7488583061542051450829,
%U A002111 1587688770629724715374457,382218817191632327375004833
%N A002111 Glaisher's G numbers.
%C A002111 Related to the formula sum(k>0,sin(kx)/k^(2n+1))=(-1)^(n+1)/2*x^(2n+1)/
               (2n+1)!*sum(i=0,2n,(2Pi/x)^i*B(i)*C(2n+1,i)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 01 2002
%C A002111 a(n)=(-1)^n(6n+3)s(2n), if n>0, where s(n) are the cubic Bernoulli numbers. 
               - Michael Somos Feb 26 2004
%D A002111 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002111 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002111 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 A002111 J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian 
               numbers, Proc. London Math. Soc., 31 (1899), 216-235.
%D A002111 S. Cooper, Cubic elliptic functions, Res. Lett. Inf. Math. Sci., Vol. 
               5, 2003, 23-59, see page 30.
%H A002111 T. D. Noe, <a href="b002111.txt">Table of n, a(n) for n = 1..50</a>
%H A002111 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A002111 <a href="Sindx_Ge.html#Glaisher">Index entries for sequences related 
               to Glaisher's numbers</a>
%H A002111 S. Cooper, <a href="http://www.massey.ac.nz/~wwiims/research/letters/
               volume5/04cooper.pdf">Cubic elliptic functions</a>.
%F A002111 To get these numbers, expand the e.g.f. (3/2)/(1+exp(x)+exp(-x)), multiply 
               coefficient of x^n by (n+1)! and take absolute values.
%F A002111 Or expand the e.g.f. (3/2)/(1+2*cos(x)) and multiply coefficient of x^n 
               by (n+1)!. - Herb Conn, Feb 25 2002
%F A002111 (2n+1)*I(n), where I(n) is given by A047788/A047789.
%F A002111 a(n)=sum(i=0, 2n, B(i)*C(2n+1, i)*3^i) where B(i) are the Bernoulli numbers, 
               C(2n, i) the binomial numbers. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 01 2002
%F A002111 E.g.f.: 3x/(2+4cos(x))-x/2 = Sum_{n>=0} a(n)x^(2n+1)/(2n+1)!. - Michael 
               Somos Feb 26 2004
%p A002111 read transforms; t1 := (3/2)/(1+exp(x)+exp(-x)); series(t1,x,50): t2 
               := SERIESTOLISTMULT(t1); [seq(n*t2[n],n=1..nops(t5))];
%o A002111 (PARI) a(n)=if(n<1,0,n*=2;(n+1)!*polcoeff(3/(2+4*cos(x+O(x^n))),n)) - 
               Michael Somos Feb 26 2004
%o A002111 (PARI) a(n)=if(n<1,0,-(-1)^n*sum(i=0,2*n,binomial(2*n+1,i)*bernfrac(i)*3^i)) 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), May 01 2002
%Y A002111 Sequence in context: A145088 A062995 A104600 this_sequence A001819 A064618 
               A075986
%Y A002111 Adjacent sequences: A002108 A002109 A002110 this_sequence A002112 A002113 
               A002114
%K A002111 nonn,nice,easy
%O A002111 1,2
%A A002111 N. J. A. Sloane (njas(AT)research.att.com).

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