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A002288 G.f.: q * Product_{m>=1} (1-q^m)^8*(1-q^2m)^8.
(Formerly M4483 N1898)
9
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.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.

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 binom(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.

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.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Index entries for sequences mentioned by Glaisher

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-Fran├žois 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: A069186 A166625 A038290 * A216711 A137232 A147764

Adjacent sequences:  A002285 A002286 A002287 * A002289 A002290 A002291

KEYWORD

sign,easy,nice,mult

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Extended, and better description added by N. J. A. Sloane, Jan 15 1996

STATUS

approved

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Last modified October 21 14:04 EDT 2019. Contains 328299 sequences. (Running on oeis4.)