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A002171 Glaisher's chi numbers. a(n) = chi(4*n + 1).
(Formerly M0745 N0280)
17
1, -2, -3, 6, 2, 0, -1, -10, 0, -2, 10, 6, -7, 14, 0, -10, -12, 0, -6, 0, 9, -4, 10, 0, 18, -2, 0, 6, -14, -18, -11, 12, 0, 0, -22, 0, 20, 14, -6, 22, 0, 0, 23, -26, 0, -18, 4, 0, -14, -2, 0, -20, 0, 0, 0, 12, 3, 30, 26, 0, -30, 14, 0, 0, 2, 30, -28, -26, 0, -18, 10, 0, -13, -34, 0, 0, 20, 0, 26, 22, 0, -6, 0, 6, 18, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number 49 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).
Glaisher (1884) essentially defines chi(n) as the sum over all solutions of n = x^2 + y^2 with even y and nonnegative odd x of x * (-1)^((x + y - 1)/2) and proves that it is multiplicative. If n is not == 1 (mod 4) then chi(n) = 0. - Michael Somos, Jun 18 2012
Denoted by g_2(q) in Cynk and Hulek on page 8 as the unique weight 2 level 32 newform. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - Michael Somos, Aug 24 2012
The weight 2 level N = 32 newform (eta(q^4)*eta(q^8))^2 belongs to the elliptic curves y^2 = x^3 + 4*x , y^2 = x^3 - x, y^2 = x^3 - 11*x - 14 and y^2 = x^3 - 11*x + 14. See the Martin-Ono link, Theorem 2, row N = 32, and the Cremona link, Table 1, N = 32. - Wolfdieter Lang, Dec 26 2016
REFERENCES
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 1001 terms from T. D. Noe)
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
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.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (psi(x) * phi(-x))^2 = phi(-x) * f(-x^2)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q)^2 * eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [-2, -4, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) and b(p) = p - number of solutions of y^2 = x^3 - x (mod p). - Michael Somos, Jul 27 2006. b(p(n)) = A278720(n). - Wolfdieter Lang, Dec 26 2016
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(2*k)))^2.
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (Sum_{k>=0} (-1)^k * (2*k + 1) * x^(2*k + 1)^2) * (Sum_{k in Z} (-1)^k * x^(4*k)^2).
Coefficients of L-series for elliptic curve "32a2": y^2 = x^3 - x.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
EXAMPLE
G.f. = 1 - 2*x - 3*x^2 + 6*x^3 + 2*x^4 - x^6 - 10*x^7 - 2*x^9 + 10*x^10 + ...
G.f. (eta(q^4)*eta(q^8))^2 = q - 2*q^5 - 3*q^9 + 6*q^13 + 2*q^17 - q^25 - 10*q^29 - 2*q^37 + 10*q^41 + ...
MATHEMATICA
max=100; f[x_] := Product[(1-x^k)*(1-x^(2k)), {k, 1, max}]^2; CoefficientList[ Series[ f[x], {x, 0, max}], x](* Jean-François Alcover, Jan 04 2012, after g.f. *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x^2]^3, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)
PROG
(PARI) {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, -1, 0], 1), 4*n + 1))}; /* Michael Somos, Jul 27 2006 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^2, n))}; /* Michael Somos, Jul 27 2006 */
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor( 4*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==3, (-p)^(e/2) * (1 + (-1)^e) / 2, forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0 = 1; y = a1 = x * (-1)^((x + y)\2) * if(y, 2, 1); for(i=2, e, x = y * a1 - p * a0; a0=a1; a1=x); a1 )))}; /* Michael Somos, Jun 18 2012 */
(Magma) A := Basis( ModularForms( Gamma0(32), 2), 341); A[2] - 2*A[6]; /* Michael Somos, Jun 12 2014 */
(Magma) qEigenform( EllipticCurve( [0, 0, 0, -1, 0]), 341); /* Michael Somos, Jun 12 2014 */
(Magma) Basis( CuspForms( Gamma0(32), 2), 341)[1]; /* Michael Somos, Mar 25 2015 */
CROSSREFS
Sequence in context: A093095 A260611 A263502 * A138515 A107410 A132041
KEYWORD
sign,easy,nice
AUTHOR
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)