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A002288 G.f.: q * Product_{m>=1} (1-q^m)^8*(1-q^2m)^8.
(Formerly M4483 N1898)
10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
This is Glaisher's Theta(n). - N. J. A. Sloane, Nov 26 2018
Number 2 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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)
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Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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.]
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
H Movasati, Y Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411, 2016.
H.-G. Quebbemann, Lattices with theta-functions for G(sqrt(2)) and linear codes, J. Algebra, 105 (1987), 443-450.
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
EXAMPLE
G.f. = q - 8*q^2 + 12*q^3 + 64*q^4 - 210*q^5 - 96*q^6 + 1016*q^7 - 512*q^8 + ...
MAPLE
t1 := product((1-q^m)^8, m=1..40): subs(q=q^2, t1): series(q*t1*%, q, 40);
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 *)
PROG
(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
(Magma) Basis( CuspForms( Gamma0(2), 8), 100) [1]; /* Michael Somos, Dec 09 2013 */
CROSSREFS
Cf. A030211.
Sequence in context: A166625 A370651 A038290 * A216711 A137232 A147764
KEYWORD
sign,easy,nice,mult
AUTHOR
EXTENSIONS
Extended, and better description added by N. J. A. Sloane, Jan 15 1996
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)