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A008457
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a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^3.
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12
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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
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OFFSET
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1,2
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COMMENTS
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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).
a(n) = r_8(n)/16, where r_8(n) = A000143(n) is the number of integral solutions of Sum_{j=1..8} x_j^2 = n (with the order of the summands respected). See the Grosswald reference, and the Hardy reference, pp. 146-147, eq. (9.9.3) and sect. 9.10. - Wolfdieter Lang, Jan 09 2017
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REFERENCES
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Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.6).
Emil Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121, eq. (9.19).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
F. Hirzebruch, T. Berger and R. Jung, Manifolds and Modular Forms, Vieweg, 1994, pp. 77, 133.
Hans Petersson, Modulfunktionen und Quadratische Formen, Springer-Verlag, 1982; p. 179.
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LINKS
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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, Sep 10 2001
a(n) = (-1)^n*(sum of cubes of even divisors of n - sum of cubes of odd divisors of n), see A051000. Sum_{n>0} n^3*x^n*(15*x^n-(-1)^n)/(1-x^(2*n)). - Vladeta Jovovic, 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). [corrected by Vaclav Kotesovec, Sep 26 2015]
Dirichlet g.f. zeta(s)*zeta(s-3)*(1-2^(1-s)+2^(4-2s)), Dirichlet convolution of A001158 and the quasi-finite (1,-2,0,16,0,0,...). - R. J. Mathar, Mar 04 2011
G.f.: (theta_3(x)^8 - 1)/16, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
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EXAMPLE
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G.f. = 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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MATHEMATICA
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nmax = 40; Rest[CoefficientList[Series[Product[((1-(-q)^k)/(1+(-q)^k))^8, {k, 1, nmax}]/16, {q, 0, nmax}], q]] (* Vaclav Kotesovec, Sep 26 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^8 - 1) / 16, {x, 0, n}]; (* Michael Somos, Aug 10 2018 *)
f[2, e_] := (8^(e+1)-15)/7; f[p_, e_] := (p^(3*e+3)-1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)
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PROG
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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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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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