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