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 1-3 of 3 results.

A002706 Theta series of 6-dimensional lattice A_6^(2) (other names for this lattice or the corresponding quadratic form are LAMBDA_{3,lambda}, P_6^(5), phi_6, F_14).

Original entry on oeis.org

1, 0, 42, 56, 84, 168, 280, 336, 462, 336, 840, 672, 1176, 1176, 1386, 1008, 1848, 2016, 2058, 2520, 3528, 2408, 3108, 2688, 4760, 3024, 5880, 4592, 6468, 4704, 5040, 6720, 6930, 6832, 10080, 7224, 7812, 7392, 12600, 7056, 14280, 11760, 12040, 9408
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In Elkies 1999 the g.f. is denoted by theta_L. - Michael Somos, Nov 09 2014

Examples

			G.f. = 1 + 42*q^2 + 56*q^3 + 84*q^4 + 168*q^5 + 280*q^6 + 336*q^7 + 462*q^8 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Intro. to 3rd ed.
  • N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103). See page 72.

Links

Crossrefs

Cf. A002653, A002656.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(7), 3), 44); A[1] + 42*A[3] + 56*A[4] + 84*A[5] + 168*A[6] + 280*A[7];  /* Michael Somos, Nov 09 2014 */
  • Mathematica
    s = (EllipticTheta[3, 0, q] *EllipticTheta[3, 0, q^7] + EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^7])^3 - 6q*(QPochhammer[q] *QPochhammer[q^7])^3 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from first formula *)
  • PARI
    {a(n) = local(A, t1, t2, t3); if( n<1, n==0, A = x * O(x^n); t1 = x * (eta(x + A) * eta(x^7 + A))^3; t2 = sum(k=1, (sqrtint(4*n + 1)  + 1)\2, 2 * x^(k*k - k), A); t3 = sum(k=1, sqrtint(n), 2 * x^(k*k), 1 + A); A = x * O(x^(n\7)); polcoeff( (t3 * subst(t3 + A, x, x^7) + x^2 * t2 * subst(t2 + A, x, x^7))^3 - 6*t1, n))}; /* Michael Somos, Jun 03 2005 */
  • Sage
    A = ModularForms( Gamma1(7), 3, prec=25) . basis(); (-21*A[0] + 4*A[1] + 21*A[2] + 105*A[3] + 224*A[4] + 441*A[5] + 672*A[6])/4 # Michael Somos, May 25 2014

Formula

a(n) = A002653(n) - 6*A002656(n).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) is a homogeneous degree 6 polynomial with 28 terms. - Michael Somos, Jun 03 2005

A105634 Expansion of Sum_{k>0} Kronecker(k,7)*x^k*(1 + x^k)/(1 - x^k)^3.

Original entry on oeis.org

1, 5, 8, 21, 24, 40, 49, 85, 73, 120, 122, 168, 168, 245, 192, 341, 288, 365, 360, 504, 392, 610, 530, 680, 601, 840, 656, 1029, 842, 960, 960, 1365, 976, 1440, 1176, 1533, 1370, 1800, 1344, 2040, 1680, 1960, 1850, 2562, 1752, 2650, 2208, 2728, 2401, 3005
Offset: 1

Views

Author

Michael Somos, Apr 16 2005, Mar 31 2008

Keywords

Examples

			q + 5*q^2 + 8*q^3 + 21*q^4 + 24*q^5 + 40*q^6 + 49*q^7 + 85*q^8 + 73*q^9 + ...

References

  • A. Balog, H. Darmon and K. Ono, Congruence for Fourier coefficients of half-integral weight modular forms and special values of L-functions, pp. 105-128 of Analytic number theory, Vol. 1, Birkhäuser, Boston, 1996, see page 107.
  • Bruce Berndt, Commentary on Ramanujan's Papers, pp. 357-426 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea, 2000. See page 372 (4).

Crossrefs

Cf. A002656, A053724, A138809, A327135.

Programs

  • Mathematica
    f[p_, e_] := If[MemberQ[{1, 2, 4}, Mod[p, 7]], (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)]; f[7, e_] := 7^(2*e); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)
  • PARI
    {a(n)=local(A,p,e); if(n<2, n==1, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==7, p^(2*e), if(kronecker(p,7)==1, (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)))))) }
  • PARI
    {a(n)=local(A,B); if(n<1, 0, n--; A=x*O(x^n); polcoeff( if(B=eta(x^7+A), A=eta(x+A); (A*B)^3+8*x*B^7/A), n))}
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-7, n / d)))}

Formula

Multiplicative with a(p^e) = p^(2e) if p = 7; (p^(2e+2)-1)/(p^2-1) if p == 1, 2, 4 (mod 7); (p^(2e+2)+(-1)^e)/(p^2+1) if p == 3, 5, 6 (mod 7).
G.f.: Sum_{k>0} Kronecker(k, 7)*x^k*(1+x^k)/(1-x^k)^3.
a(n) = A002656(n) + 8*A053724(n-2).
a(7n) = 49a(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A138809.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 32*Pi^3/(343*sqrt(7)) = 1.093343069... (A327135). - Amiram Eldar, Nov 16 2023

A129666 Expansion of unique cusp form of weight 4 level 7 in powers of q.

Original entry on oeis.org

1, -1, -2, -7, 16, 2, -7, 15, -23, -16, -8, 14, 28, 7, -32, 41, 54, 23, -110, -112, 14, 8, 48, -30, 131, -28, 100, 49, -110, 32, 12, -161, 16, -54, -112, 161, -246, 110, -56, 240, 182, -14, 128, 56, -368, -48, 324, -82, 49, -131, -108, -196, -162, -100, -128
Offset: 1

Views

Author

Michael Somos, Apr 27 2007

Keywords

Comments

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

Examples

			G.f. = q - q^2 - 2*q^3 - 7*q^4 + 16*q^5 + 2*q^6 - 7*q^7 + 15*q^8 - 23*q^9 - ...

References

  • H. Rosson and G. Tornaria, Central values of quadratic twists for a modular form of weight 4, pp. 315-321 of J. B. Conrey et al., ed., Ranks of Elliptic Curves and Random Matrix Theory, Cambridge University Press, 2007.

Links

Crossrefs

Cf. A002652, A002656.

Programs

  • Magma
    Basis( CuspForms( Gamma0(7), 4), 56)[1]; /* Michael Somos, Nov 11 2015 */
  • Mathematica
    a[ n_] := With[ {A1 = QPochhammer[ q] QPochhammer[ q^7], A2 = QPochhammer[ q^2] QPochhammer[ q^14]}, SeriesCoefficient[  (A1^3 + 4 q A2^3) A1^2 / A2, {q, 0, n}]]; (* Michael Somos, Nov 11 2015 *)
  • PARI
    {a(n) = my(A, A1, A2); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4*x * A2^3) * A1^2 / A2, n))};
  • Sage
    CuspForms( Gamma0(7), 4, prec=55).0; # Michael Somos, May 28 2013

Formula

Expansion of q * phi(-q)^3 * psi(q) * phi(-q^7)^3 * psi(q^7) + 4*q^2 * (phi(-q) * psi(q) * phi(-q^7) * psi(q^7))^2 in powers of q.
Expansion of ((eta(q) * eta(q^7))^3 + 4 * (eta(q^2) * eta(q^14))^3) * (eta(q) * eta(q^7))^2 / (eta(q^2) * eta(q^14)) in powers of q.
a(n) is multiplicative with a(7^e) = (-7)^e, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 49 (t/i)^4 f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 + 2*u*v + 16*u*w + 12*v^2 + 32*v*w + 256*w^2) * (-v^3 + 2*w*u*v + w*u^2 + 16*w^2*u) + 2*v^5.
Convolution of A002652 and A002656.
Showing 1-3 of 3 results.

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