%I A005798 M4528
%S A005798 0,1,8,44,192,718,2400,7352,20992,56549,145008,356388,844032,1934534,
%T A005798 4306368,9337704,19771392,40965362,83207976,165944732,325393024,
%U A005798 628092832,1194744096,2241688744,4152367104,7599231223,13749863984
%V A005798 0,1,-8,44,-192,718,-2400,7352,-20992,56549,-145008,356388,-844032,1934534,
               -4306368,
%W A005798 9337704,-19771392,40965362,-83207976,165944732,-325393024,628092832,-1194744096,
%X A005798 2241688744,-4152367104,7599231223,-13749863984
%N A005798 eta(z/2)^8*eta(2z)^16/eta(z)^24, where eta = Dedekind's function.
%C A005798 Expansion of elliptic lambda/16 = m/16 = (k/4)^2 in powers of the nome 
               q.
%C A005798 Euler transform of period 4 sequence [ -8,16,-8,0,...].
%C A005798 G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v)=u^2-v+16uv-32u^2v+256(uv)^2. 
               - Michael Somos Mar 19 2004
%D A005798 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005798 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math.Series 55, Tenth Printing, 
               1972, p. 591.
%D A005798 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
%D A005798 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 
               3, p. 23.
%H A005798 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A005798 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 
               1972, p. 591.
%H A005798 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EllipticLambdaFunction.html">Elliptic Lambda Function</a> a section 
               of The World of Mathematics.
%F A005798 G.f.: q* Product( (1+q^(2n))/(1+q^(2n-1)), n=1..inf )^8 = eta(q)^8*eta(q^4)^16/
               eta(q^2)^24; eta = Dedekind's function.
%p A005798 with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; 
               local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), 
               j=1..n)/n fi end end: aa:=etr (n-> [ -8,16,-8,0] [modp(n-1,4)+1]): 
               a:= n->aa(n-1): seq (a(n), n=0..26); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Sep 08 2008]
%o A005798 (PARI) a(n)=local(A,m); if(n<1,0,m=1; A=x+O(x^2); while(m<n,m*=2; A=sqrt(subst(A,
               x,x^2)); A=A/(1+4*A)^2); polcoeff(A,n))
%o A005798 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^4+A)^2/
               eta(x^2+A)^3)^8, n))} /* Michael Somos Jul 16 2005 */
%Y A005798 If initial 0 is omitted and sequence begins with a(0) = 1, then this 
               is the convolution of A001938 with itself. G.f.s are related by A005798(x)=x*A001938(x)^2. 
               Reversion of A005797. Cf. A007248, A029845.
%Y A005798 Sequence in context: A000938 A165618 A059596 this_sequence A092877 A023007 
               A073380
%Y A005798 Adjacent sequences: A005795 A005796 A005797 this_sequence A005799 A005800 
               A005801
%K A005798 sign,easy
%O A005798 0,3
%A A005798 N. J. A. Sloane (njas(AT)research.att.com).