A279955 -id:A279955 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

			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 + ...

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).

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)
J. E. Cremona, Algorithms for Modular Elliptic Curves.
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]
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.
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.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to Glaisher's numbers

Cf. A000203, A002172, A278720, A279955.

A := Basis( ModularForms( Gamma0(32), 2), 341); A[2] - 2*A[6]; /* Michael Somos, Jun 12 2014 */

qEigenform( EllipticCurve( [0, 0, 0, -1, 0]), 341); /* Michael Somos, Jun 12 2014 */

Basis( CuspForms( Gamma0(32), 2), 341)[1]; /* Michael Somos, Mar 25 2015 */

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 *)

{a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, -1, 0], 1), 4*n + 1))}; /* Michael Somos, Jul 27 2006 */

{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 */

{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 */

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

A280339 Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -1, -2, -5, -14, 4, 12, 5, 40, 0, -26, 11, -68, -15, 30, -18, 106, 3, -50, -10, -182, 29, 104, 10, 270, 11, -130, 37, -360, -51, 164, -16, 506, -30, -266, -65, -686, 62, 320, 53, 898, 22, -414, 50, -1206, -61, 612, -52, 1560, -4, -696, -81, -1958, 120

Offset: 0

Michael Somos, Dec 31 2016

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 + 4*x^6 + 12*x^7 + 5*x^8 + ...
G.f. = q^-1 + 2*q^3 - q^7 - 2*q^11 - 5*q^15 - 14*q^19 + 4*q^23 + 12*q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A138515, A279955.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 QPochhammer[ x]^2 QPochhammer[ -x^2, x^4]^4, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^4 + A)^4 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};

Expansion of phi(-x^4)^2 * chi(-x^4)^2 * f(x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
Expansion of q^(1/4) * eta(q^2)^6 * eta(q^4)^4 / (eta(q)^2 * eta(q^8)^4) in powers of q.
Euler transform of period 8 sequence [2, -4, 2, -8, 2, -4, 2, -4, ...].
a(n) = (-1)^n * A279955(n).
a(3*n + 1) / a(1) == A138515(n) (mod 3). a(3^3*n + 7) / a(7) == A138515(n) (mod 3^2).

A275372 Expansion of f(-x) * f(-x^2)^4 / phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, -7, 6, 20, -13, -34, 15, 53, -25, -91, 52, 135, -65, -180, 82, 253, -133, -343, 160, 449, -207, -603, 306, 780, -348, -979, 438, 1241, -600, -1557, 703, 1924, -890, -2375, 1115, 2910, -1300, -3535, 1620, 4318, -1993, -5198, 2335, 6180, -2783, -7420

Offset: 0

Michael Somos, Dec 25 2016

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x - 7*x^2 + 6*x^3 + 20*x^4 - 13*x^5 - 34*x^6 + 15*x^7 + 53*x^8 + ...
G.f. = q^3 - q^11 - 7*q^19 + 6*q^27 + 20*q^35 - 13*q^43 - 34*q^51 + 15*q^59 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A279955.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2]^4 / EllipticTheta[ 3, 0, x^2], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^7 / (QPochhammer[ -x] QPochhammer[ -x^2]^2), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^6 * eta(x^8 + A)^2 / eta(x^4 + A)^5, n))};

Expansion of f(-x^2)^7 / (f(x) * f(x^2)^2) in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-3/8) * eta(q) * eta(q^2)^6 * eta(q^8)^2 / eta(q^4)^5 in powers of q.
Euler transform of period 8 sequence [ -1, -7, -1, -2, -1, -7, -1, -4, ...].
G.f.: Product_{k>0} (1 - x^k)^4 * (1 + x^k)^3 * (1 + x^(4*k))^2 / (1 + x^(2*k))^3.
2 * a(n) = - A279955(2*n + 1).

A279918 Expansion of f(-x^2)^7 / (f(x) * f(-x^8)^2) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, -5, 4, 5, 0, 11, -15, -18, 3, -10, 29, 10, 11, 37, -51, -16, -30, -65, 62, 53, 22, 50, -61, -52, -4, -81, 120, 62, 0, 124, -182, -85, -43, -157, 171, 123, 60, 202, -198, -174, 0, -190, 301, 117, 54, 278, -375, -171, -153, -399, 370, 300, 108, 408, -451

Offset: 0

Michael Somos, Dec 23 2016

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x - 5*x^2 + 4*x^3 + 5*x^4 + 11*x^6 - 15*x^7 - 18*x^8 + ...
G.f. = q^-1 - q^7 - 5*q^15 + 4*q^23 + 5*q^31 + 11*q^47 - 15*q^55 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A279955.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^3 QPochhammer[ -x^2, x^4]^2 QPochhammer[ x]^4, {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2]^4 QPochhammer[ -x^2, x^4]^2 QPochhammer[ x, -x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^4 * eta(x^4 + A) / eta(x^8 + A)^2, n))};

lista(nn) = {q='q+O('q^nn); Vec(eta(q)*eta(q^2)^4*eta(q^4)/eta(q^8)^2)} \\ Altug Alkan, Mar 21 2018

Expansion of chi(x)^3 * chi(x^2)^2 * f(-x)^4 in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of phi(-x^2)^4 * chi(x^2)^2 / chi(x) in powers of x where chi(), phi() are Ramanujan theta functions.
Expansion of q^(-1/8) * eta(q) * eta(q^2)^4 * eta(q^4) / eta(q^8)^2 in powers of q.
Euler transform of period 8 sequence [ -1, -5, -1, -6, -1, -5, -1, -4, ...].
a(n) = A279955(2*n).

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

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A280339 Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

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A275372 Expansion of f(-x) * f(-x^2)^4 / phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

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A279918 Expansion of f(-x^2)^7 / (f(x) * f(-x^8)^2) in powers of x where f() is a Ramanujan theta function.

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