cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A006571 Expansion of q*Product_{k>=1} (1-q^k)^2*(1-q^(11*k))^2.

Original entry on oeis.org

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2, 2, -2, -1, 0, -4, -8, 5, -4, 0, 2, 7, 8, -1, 4, -2, -4, 3, 0, -4, 0, -8, -4, -6, 2, -2, 2, 8, 4, -3, 8, 2, 8, -6, -10, 1, 0, 0, 0, 5, -2, 12, -14, 4, -8, 4, 2, -7, -4, 1, 4, -3, 0, 4, -6, 4, 0, -2, 8, -10, -4, 1, 16, -6, 4, -2, 12, 0, 0, 15, 4, -8, -2, -7, -16, 0, -8, -7, 6, -2, -8
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number 23 of the 74 eta-quotients listed in Table I of Martin (1996).
Unique cusp form of weight 2 for congruence group Gamma_1(11). - Michael Somos, Aug 11 2011
For some elliptic curves with p-defects given by this sequence, and for more references, see A272196. See also the Michael Somos formula from May 23 2008 below. - Wolfdieter Lang, Apr 25 2016

Examples

			G.f.: q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + ...

References

  • Barry Cipra, What's Happening in the Mathematical Sciences, Vol. 5, Amer. Math. Soc., 2002; see p. 5.
  • M. du Sautoy, Review of "Love and Math: The Heart of Hidden Reality" by Edward Frenkel, Nature, 502 (Oct 03 2013), p. 36.
  • N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 42.
  • J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 412.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • A. Wiles, Modular forms, elliptic curves and Fermat's last theorem, pp. 243-245 of Proc. Intern. Congr. Math. (Zurich), Vol. 1, 1994.

Links

Crossrefs

Cf. A002070 (terms with prime indices), A032442, A030200.

Programs

  • Magma
    [ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1),n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 29 2007 */
  • Magma
    [ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 31 2007 */
  • Magma
    Basis( CuspForms( Gamma1(11), 2), 101) [1]; /* Michael Somos, Jul 14 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ q (Product[ (1 - q^k), {k, 11, n, 11}] Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
  • PARI
    {a(n) = if( n<1, 0, ellak( ellinit( [0, -1, 1, 0, 0], 1), n))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};
  • PARI
    {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, 1, a0=1; a1 = y = -sum(x=0, p-1, kronecker( 4*x^3 - 4*x^2 + 1, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */
  • Sage
    CuspForms( Gamma1(11), 2, prec = 101).0 # Michael Somos, Aug 11 2011

Formula

Expansion of (eta(q) * eta(q^11))^2 in powers of q.
a(n) == A000594(n) (mod 11). [Cowles]. - R. J. Mathar, Feb 13 2007
Euler transform of period 11 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...]. - Michael Somos, Feb 12 2006
a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) for p != 11. - Michael Somos, Feb 12 2006
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u*w * (u + 4*v + 4*w) - v^3. - Michael Somos, Mar 21 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11 (t/i)^2 f(t) where q = exp(2 Pi i t).
Convolution square of A030200.
Coefficients of L-series for elliptic curve "11a3": y^2 + y = x^3 - x^2. - Michael Somos, May 23 2008
Convolution inverse is A032442. - Michael Somos, Apr 21 2015
a(prime(n)) = prime(n) - A272196(n), n >= 3.
a(2) = -2 is not 2 - A272196(1) = 0. Modularity pattern of some elliptic curves. - Wolfdieter Lang, Apr 25 2016

A129522 Expansion of unique weight 3 level 11 multiplicative cusp form in powers of q.

Original entry on oeis.org

1, 0, -5, 4, -1, 0, 0, 0, 16, 0, -11, -20, 0, 0, 5, 16, 0, 0, 0, -4, 0, 0, 35, 0, -24, 0, -35, 0, 0, 0, -37, 0, 55, 0, 0, 64, -25, 0, 0, 0, 0, 0, 0, -44, -16, 0, 50, -80, 49, 0, 0, 0, -70, 0, 11, 0, 0, 0, 107, 20, 0, 0, 0, 64, 0, 0, 35, 0, -175, 0, -133, 0, 0
Offset: 1

Views

Author

Michael Somos, Apr 19 2007, Jun 06 2007

Keywords

Comments

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661.

Examples

			G.f. = q - 5*q^3 + 4*q^4 - q^5 + 16*q^9 - 11*q^11 - 20*q^12 + 5*q^15 + 16*q^16 + ...

Crossrefs

Cf. A006571, A028609, A030200, A035179, A065099, A138661.

Programs

  • Magma
    A := Basis( CuspForms( Gamma1(11), 3), 73); A[1] - 5*A[3] + 4*A[4] - A[5]; /* Michael Somos, Mar 26 2015 */
  • Mathematica
    a[ n_] := Module[ {A, B}, B = QPochhammer[ q] QPochhammer[ q^11]; A = B / (q QPochhammer[ q^3] QPochhammer[ q^33]); SeriesCoefficient[ q B^3 (1 + 3 / A) Sqrt[ q (A + 1 + 3 / A)], {q, 0, n}]]; (* Michael Somos, Mar 26 2015 *)
  • PARI
    {a(n) = my(A, B); if( n<1, 0, n--; A = x * O(x^n); B = eta(x + A) * eta(x^11 + A); A = B /( x * eta(x^3 + A) * eta(x^33 + A)); A = B^3 * (1 + 3/A) * sqrt(x * (A + 1 + 3/A)); polcoeff(A, n))};
  • PARI
    {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, (-11)^e, kronecker( -11, p)==-1, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\11), if( issquare(4*p - 11*x^2, &y), break)); y = y^2 - p*2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 06 2007 */
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); A = eta(x + A) * eta(x^11 + A); polcoeff( A^2 / subst(A + x * O(x^(n\2)), x, x^2) * (A^2 + 4*x * subst(A + x * O(x^(n\2)), x, x^2)^2 + 8 * x^3 * subst(A + x * O(x^(n\4)), x, x^4)^2), n))}; /* Michael Somos, Jun 06 2007 */

Formula

Expansion of (F(q)^2 + 4*F(q^2)^2 + 8*F(q^4)^2) * F(q)^2 / F(q^2) in powers of q where F(q) := eta(q) * eta(q^11) is the g.f. of A030200.
a(n) is multiplicative with a(11^e) = (-11)^e, a(p^e) = (1+(-1)^e)/2*p^e if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^2 - 2*p and 4*p = y^2 + 11*x^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
G.f.: (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * x^(u*u + u*v + 3*v*v). - Michael Somos, Jun 14 2007
Convolution of A006571 and A028609. - Michael Somos, Aug 14 2012
a(4*n + 2) = 0. - Michael Somos, Nov 11 2015

A208664 Expansion of f(x) * f(x^11) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, -1, 0, -1, 0, 0, 0, 0, -2, 1, 0, 2, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, -2, 0, 0, 0, -1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, -1, 0, -2, 0, 0, 1, 0, 0, 0, -1, -1, 2
Offset: 0

Views

Author

Michael Somos, Mar 05 2012

Keywords

Comments

Number 69 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).

Examples

			G.f. = 1 + x - x^2 - x^5 - x^7 + x^11 - x^13 + x^15 - x^16 - x^18 - 2*x^23 + ...
G.f. = q + q^3 - q^5 - q^11 - q^15 + q^23 - q^27 + q^31 - q^33 - q^37 - 2*q^47 + ...

Links

Crossrefs

Cf. A030200.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^11] , {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
  • PARI
    {a(n) = my(A, p, e, f); if(n<0, 0, n = 2*n+1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==11, (-1)^e, f = sum( k=0, p-1, (k^3-k^2-k-1)%p == 0); (-1)^(e*(p-1)/2) * if( f==0, (e-1)%3-1, f==1, (1 + (-1)^e) / 2, e+1))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^22 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^11 + A) * eta(x^44 + A)), n))};

Formula

Expansion of q^(-1/2) * eta(q^2)^3 * eta(q^22)^3 / (eta(q) * eta(q^4) * eta(q^11) * eta(q^44)) in powers of q.
Euler transform of period 44 sequence [ 1, -2, 1, -1, 1, -2, 1, -1, 1, -2, 2, -1, 1, -2, 1, -1, 1, -2, 1, -1, 1, -4, 1, -1, 1, -2, 1, -1, 1, -2, 1, -1, 2, -2, 1, -1, 1, -2, 1, -1, 1, -2, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (176 t)) = 176^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = (-1)^e, b(p^e) = s * ((e-1)%3 - 1) if f=0, b(p^e) = s * (e + 1) if f=3, b(p^e) = s * (1 + (-1)^e) / 2 if f=1 where s = (-1)^(e*(p-1)/2) and f = number of zeros of x^3-x^2-x-1 modulo p.
a(n) = (-1)^n * A030200(n). a(11*n + 3) = a(11*n + 6) = a(11*n + 8) = a(11*n + 9) = a(11*n + 10) = 0. a(11*n + 5) = -a(n).

A252731 Fourier expansion of the unique newform on Gamma_0(44).

Original entry on oeis.org

1, 1, -3, 2, -2, -1, -4, -3, 6, 8, 2, -3, 4, -5, 0, 5, -1, -6, -1, -4, 0, -10, 6, 0, -3, 6, -6, 3, 8, 3, -4, -4, 12, -1, -3, 15, -4, 4, -2, 2, 1, 6, -18, 0, -9, -8, 5, -24, -7, 2, 18, 8, -6, 6, 2, -1, -15, 9, 8, 12, 1, 0, 3, -16, -10, -6, 16, 15, 9, 14, 0, 4
Offset: 0

Views

Author

Michael Somos, Dec 21 2014

Keywords

Comments

Fourier expansion denoted by f_44(tau) on p. 80 of Umbral Moonshine.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x - 3*x^2 + 2*x^3 - 2*x^4 - x^5 - 4*x^6 - 3*x^7 + 6*x^8 + ...
G.f. = q + q^3 - 3*q^5 + 2*q^7 - 2*q^9 - q^11 - 4*q^13 - 3*q^15 + 6*q^17 + ...

Links

Crossrefs

Cf. A030200.

Programs

  • Magma
    A := Basis( CuspForms( Gamma0(44), 2), 146); A[1] + A[3];
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11] (2 EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^11] - EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^11] - 4 x QPochhammer[ x^2] QPochhammer[ x^22]), {x, 0, n}];
  • PARI
    {a(n) = local(A, F1, F2, F4); if( n<0, 0, A = x * O(x^n); F1 = eta(x + A) * eta(x^11 + A); F2 = subst(F1, x, x^2); F4 = subst(F1, x, x^4); polcoeff( 2*F2^5 / (F1 * F4^2) - F1^3 / F2 - 4*x * F1*F2, n))};
  • Sage
    A = CuspForms( Gamma0(44), 2, prec=146) . basis(); A[0] + A[2];

Formula

Expansion of 2*F(x^2)^5 / (F(x) * F(x^4)^2) - F(x)^3 / F(x^2) - 4*x * F(x) * F(x^2) in powers of x where F() is the g.f. for A030200.
Expansion of f(-q^22) * phi(-q^22)^3 / chi(-q^2) - 2 * q^4 * f(-q^22) * (chi(-q) * psi(q^11)^3 + chi(q) * psi(-q^11)^3) in powers of q^2 where phi(), psi(), chi(), f() are Ramanujan theta functions.
a(n) = b(2*n + 1) where b() is multiplicative with b(p^e) = (-1)^e if p = 11, b(p^e) = b(p)*b(p^(e-1)) - p*b(p^(e-2)) if p != 11.
G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 44 (t/i)^2 f(t) where q = exp(2 Pi i t).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.