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A002171 Glaisher's chi numbers. a(n) = chi(4*n + 1).

%I A002171 M0745 N0280 #96 Mar 28 2025 17:32:08
%S A002171 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,
%T A002171 -2,0,6,-14,-18,-11,12,0,0,-22,0,20,14,-6,22,0,0,23,-26,0,-18,4,0,-14,
%U A002171 -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
%N A002171 Glaisher's chi numbers. a(n) = chi(4*n + 1).
%C A002171 Number 49 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A002171 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A002171 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
%C A002171 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
%C A002171 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
%C A002171 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
%D A002171 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002171 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002171 Seiichi Manyama, <a href="/A002171/b002171.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H A002171 Amanda Clemm, <a href="http://www.mdpi.com/2227-7390/4/1/5">Modular Forms and Weierstrass Mock Modular Forms</a>, Mathematics, volume 4, issue 1, (2016)
%H A002171 J. E. Cremona, <a href="https://homepages.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html">Algorithms for Modular Elliptic Curves</a>.
%H A002171 S. Cynk and K. Hulek, <a href="http://arXiv.org/abs/math/0509424">Construction and examples of higher-dimensional modular Calabi-Yau manifolds</a>, arXiv:math/0509424 [math.AG], 2005-2006.
%H A002171 S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H A002171 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
%H A002171 J. W. L. Glaisher, <a href="http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=PPN600494829_0020%7CLOG_0017">On the function chi(n)</a>, Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
%H A002171 J. W. L. Glaisher, <a href="/A002171/a002171.pdf">On the function chi(n)</a>, Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
%H A002171 T. Ishikawa, <a href="http://projecteuclid.org/euclid.hmj/1206128505">Congruences between binomial coefficients binomial(2f,f) and Fourier coefficients of certain eta-products</a>, Hiroshima Math. J. 22 (1992), no. 3, 583-590.
%H A002171 M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H A002171 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A002171 Yves Martin and Ken Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
%H A002171 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A002171 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A002171 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A002171 <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>
%F A002171 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.
%F A002171 Expansion of q^(-1/4) * eta(q)^2 * eta(q^2)^2 in powers of q.
%F A002171 Euler transform of period 2 sequence [-2, -4, ...].
%F A002171 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
%F A002171 G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(2*k)))^2.
%F A002171 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).
%F A002171 Coefficients of L-series for elliptic curve "32a2": y^2 = x^3 - x.
%F A002171 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).
%F A002171 G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - _Ilya Gutkovskiy_, Sep 19 2018
%e A002171 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 + ...
%e A002171 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 + ...
%t A002171 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. *)
%t A002171 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^2, {x, 0, n}]; (* _Michael Somos_, Jun 18 2012 *)
%t A002171 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x^2]^3, {x, 0, n}]; (* _Michael Somos_, Jun 18 2012 *)
%o A002171 (PARI) {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, -1, 0], 1), 4*n + 1))}; /* _Michael Somos_, Jul 27 2006 */
%o A002171 (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 */
%o A002171 (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 */
%o A002171 (Magma) A := Basis( ModularForms( Gamma0(32), 2), 341); A[2] - 2*A[6]; /* _Michael Somos_, Jun 12 2014 */
%o A002171 (Magma) qEigenform( EllipticCurve( [0, 0, 0, -1, 0]), 341); /* _Michael Somos_, Jun 12 2014 */
%o A002171 (Magma) Basis( CuspForms( Gamma0(32), 2), 341)[1]; /* _Michael Somos_, Mar 25 2015 */
%Y A002171 Cf. A000203, A002172, A278720, A279955.
%K A002171 sign,easy,nice
%O A002171 0,2
%A A002171 _N. J. A. Sloane_