A002288 G.f.: q * Product_{m>=1} (1-q^m)^8*(1-q^2m)^8.
0, 1, -8, 12, 64, -210, -96, 1016, -512, -2043, 1680, 1092, 768, 1382, -8128, -2520, 4096, 14706, 16344, -39940, -13440, 12192, -8736, 68712, -6144, -34025, -11056, -50760, 65024, -102570, 20160, 227552, -32768, 13104, -117648, -213360, -130752, 160526, 319520
Offset: 0
Examples
G.f. = q - 8*q^2 + 12*q^3 + 64*q^4 - 210*q^5 - 96*q^6 + 1016*q^7 - 512*q^8 + ...
References
- J. W. L. Glaisher, On the representation of a number as a sum of 14 and 16 squares, Quart. J. Math. 38 (1907), 178-236 (see p. 198).
- F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.
- G. Shimura, Modular forms of half-integral weight, pp. 57-74 of Modular Functions of One Variable I (Antwerp 1972), Lect. Notes Math. 320 (1973).
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1002 terms from T. D. Noe)
- T. Ishikawa, Congruences between binomial coefficients binomial(2f,f) and Fourier coefficients of certain eta-products, Hiroshima Math. J. 22 (1992), no. 3, 583-590.
- M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
- Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
- LMFDB, Newform orbit 2.8.a.a.
- Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
- H Movasati and Y Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2021.
- H.-G. Quebbemann, Lattices with theta-functions for G(sqrt(2)) and linear codes, J. Algebra, 105 (1987), 443-450.
- Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
- Index entries for sequences mentioned by Glaisher
Crossrefs
Cf. A030211.
Programs
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Magma
Basis( CuspForms( Gamma0(2), 8), 100) [1]; /* Michael Somos, Dec 09 2013 */
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Maple
t1 := product((1-q^m)^8,m=1..40): subs(q=q^2,t1): series(q*t1*%,q,40);
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Mathematica
max = 36; f[q_] := q*Product[(1-q^m)^8*(1-q^(2m))^8, {m, 1, max}]; CoefficientList[ Series[f[q], {q, 0, max}], q] (* Jean-François Alcover, Jul 18 2012 *) a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^2])^8, {q, 0, n}]; (* Michael Somos, Apr 09 2013 *) a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}]; (* Michael Somos, Dec 09 2013 *) -
PARI
{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */ -
PARI
q='q+O('q^50); concat(0, Vec((eta(q)*eta(q^2))^8)) \\ Altug Alkan, Sep 19 2018 -
Sage
CuspForms( Gamma0(2), 8, prec=100).0; # Michael Somos, May 28 2013
Formula
Expansion of cusp form (e(1)-e(2))(e(1)-e(3))(e(2)-e(3))^2 for GAMMA_0(2).
Expansion of q * psi(q)^8 * phi(-q)^8 in powers of q where psi(), phi() are Ramanujan theta functions. - Michael Somos, Dec 09 2013
Expansion of (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Mar 18 2003
Euler transform of period 2 sequence [ -8, -16, ... ].
a(n) is multiplicative with a(2^e) = (-8)^e, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)). - Michael Somos, Mar 08 2006
Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = A2^3 + 2 * A0 * (A1^2 + A3^2) - 4 * A1*A2*A3 - 3 * A0^2*A2. - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 09 2013
a(2*n) = -8 * a(n). Convolution square of A030211. - Michael Somos, Apr 09 2013
G.f.: x*exp(8*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
Extensions
Extended, and better description added by N. J. A. Sloane, Jan 15 1996
Comments