Search: id:A008457
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%I A008457
%S A008457 1,7,28,71,126,196,344,583,757,882,1332,1988,2198,2408,3528,4679,4914,
%T A008457 5299,6860,8946,9632,9324,12168,16324,15751,15386,20440,24424,24390,
%U A008457 24696,29792,37447,37296,34398,43344,53747,50654,48020,61544,73458
%N A008457 Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^3.
%C A008457 The modular form (e(1)-e(2))(e(1)-e(3)) for GAMMA_0 (2) (with constant 
               term -1/16 omitted).
%D A008457 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 77, Eq. (31.6).
%D A008457 F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 
               133.
%D A008457 M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
%D A008457 H. Petersson, Modulfunktionen und Quadratische Formen, Springer-Verlag, 
               1982; p. 179.
%F A008457 Multiplicative with a(2^e) = (8^(e+1)-15)/7, a(p^e) = (p^(3*e+3)-1)/(p^3-1), 
               p > 2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 10 2001
%F A008457 a(n) = (-1)^n*(sum of cubes of even divisors of n - sum of cubes of odd 
               divisors of n). Sum_{n>0} n^3*x^n*(15*x^n-(-1)^n)/(1-x^(2*n)). - 
               Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 24 2002
%F A008457 G.f.: Sum_{k>0} k^3 x^k/(1-(-x)^k) . - Michael Somos Sep 25 2005
%F A008457 G.f.: (1/16)*(-1+(Product_{k>0} (1+q^k)/(1-q^k))^8).
%e A008457 q + 7*q^2 + 28*q^3 + 71*q^4 + 126*q^5 + 196*q^6 + 344*q^7 + 583*q^8 + 
               ...
%p A008457 (1/16)*product((1+q^n)^8/(1-q^n)^8,n=1..60);
%o A008457 (PARI) a(n)=if(n<1, 0, (-1)^n*sumdiv(n,d,(-1)^d*d^3)) /* Michael Somos 
               Sep 25 2005 */
%Y A008457 Cf. A000143, A064027, A002129, A048272.
%Y A008457 A138503(n) = -(-1)^n * a(n).
%Y A008457 Sequence in context: A045551 A024844 A033582 this_sequence A138503 A064951 
               A073995
%Y A008457 Adjacent sequences: A008454 A008455 A008456 this_sequence A008458 A008459 
               A008460
%K A008457 nonn,mult
%O A008457 1,2
%A A008457 N. J. A. Sloane (njas(AT)research.att.com).

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