Search: id:A000377
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%I A000377
%S A000377 1,1,1,1,1,2,1,2,1,1,2,2,1,0,2,2,1,0,1,0,2,2,2,0,1,3,0,1,2,2,2,2,1,2,
%T A000377 0,4,1,0,0,0,2,0,2,0,2,2,0,0,1,3,3,0,0,2,1,4,2,0,2,2,2,0,2,2,1,0,2,0,
%U A000377 0,0,4,0,1,2,0,3,0,4,0,2,2,1,0,2,2,0,0,2,2,0,2,0,0,2,0,0,1,2,3,2,3,2
%N A000377 Sum over divisors d of n of Kronecker symbol (-6,d), with a(0)=1.
%D A000377 G. E. Andrews, editor, P. A. MacMahon: Collected Papers Volume II, MIT 
               Press, 1986, p. 260.
%D A000377 G. E. Andrews, "Nathan Fine 1916-1994", Notices Amer. Math. Soc., 42 
               (No. 6, 1995), 678-679.
%D A000377 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 81, Eq. (32.5).
%H A000377 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A000377 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FinesEquation.html">Fine's Equation</a>
%H A000377 A. Berkovich and H. Yesilyurt, <a href="http://arXiv.org/abs/math.NT/
               0611300">Ramanujan's identities and representation of integers by 
               certain binary and quaternary quadratic forms</a>
%F A000377 Has a nice Dirichlet series expansion, see PARI line.
%F A000377 Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1, 5, 7, 
               11 (mod 24), a(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24). 
               - Michael Somos Jun 17 2005
%F A000377 G.f.: Product_{k>0} (1+x^k)(1-x^(3k))(1-x^(8k))/(1+x^(12k)).
%F A000377 G.f.: 1 + Sum_{k>0} x^k(1+x^(4k))(1+x^(6k))/(1+x^(12k)) . - Michael Somos 
               Sep 10 2005
%F A000377 Moebius transform is period 24 sequence [1,0,0,0,1,0,1,0,0,0,1,0,-1,0,
               0,0,-1,0,-1,0,0,0,-1,0,...]. - Michael Somos Jan 26 2006
%F A000377 Expansion of (phi(q)phi(q^6)+phi(q^2)phi(q^3))/2 where phi() is a Ramanujan 
               theta function. - Michael Somos Jan 26 2006
%F A000377 Expansion of eta(q^2)eta(q^3)eta(q^8)eta(q^12)/(eta(q)eta(q^24)) in powers 
               of q.
%F A000377 Euler transform of period 24 sequence [1,0,0,0,1,-1,1,-1,0,0,1,-2,1,0,
               0,-1,1,-1,1,0,0,0,1,-2,...].
%F A000377 G.f.: 1 + Sum{n = -infinity...infinity} (q^n + q^(5n)) / (1 + q^(12n)) 
               (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007
%o A000377 (PARI) a(n)=if(n<1,n==0,sumdiv(n,d,kronecker(-6,d)))
%o A000377 (PARI) a(n)=if(n<1, n==0, direuler(p=2,n, 1/(1-X)/(1-kronecker(-6,p)*X))[n])
%o A000377 (PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x^2+A)*eta(x^3+A)*eta(x^8+A)*eta(x^12+A)/
               (eta(x+A)*eta(x^24+A)), n))
%Y A000377 Sequence in context: A029439 A075117 A029810 this_sequence A115660 A128581 
               A026517
%Y A000377 Adjacent sequences: A000374 A000375 A000376 this_sequence A000378 A000379 
               A000380
%K A000377 nonn,easy,nice,mult
%O A000377 0,6
%A A000377 N. J. A. Sloane (njas(AT)research.att.com).
%E A000377 Edited by Michael Somos, Sep 10, 2002.

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