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.

A005932 Erroneous version of A003781.

Original entry on oeis.org

1, 0, 0, 56, 14, 0, 0, 576, 84, 0, 0, 1512, 280, 0, 0, 4032, 574, 0, 0, 5544, 840, 0, 0
Offset: 0

Views

Author

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A004008 Expansion of theta series of E_7 lattice in powers of q^2.

Original entry on oeis.org

1, 126, 756, 2072, 4158, 7560, 11592, 16704, 24948, 31878, 39816, 55944, 66584, 76104, 99792, 116928, 133182, 160272, 177660, 205128, 249480, 265104, 281736, 350784, 382536, 390726, 470232, 505568, 532800, 615384, 640080, 701568, 799092, 809424, 853776
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			G.f. = 1 + 126*x + 756*x^2 + 2072*x^3 + 4158*x^4 + 7560*x^5 + 11592*x^6 + ...
G.f. = 1 + 126*q^2 + 756*q^4 + 2072*q^6 + 4158*q^8 + 7560*q^10 + 11592*q^12 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125. Equation (112)
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003781, A005875, A228746.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(4), 7/2), 50); A[1] + 126*A[2]; /* Michael Somos, Jun 09 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( 8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}]; (* Michael Somos, Aug 27 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 2, 0, q]^4), {q, 0, n}]; (* Michael Somos, Apr 21 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A^3 * (8 * A^4 - 7 * subst(A, x, -x)^4), n))}; /* Michael Somos, Oct 24 2006 */
  • PARI
    {a(n) = my(G); if( n<1, n==0, G = [2, -1, 0, 0, 0, 0, 0; -1, 2, -1, 0, 0, 0, 0; 0, -1, 2, -1, 0, 0, 0; 0, 0, -1, 2, -1, 0, -1; 0, 0, 0, -1, 2, -1, 0; 0, 0, 0, 0, -1, 2, 0; 0, 0, 0, -1, 0, 0, 2]; 2 * qfrep( G, n, 1)[n])}; /* Michael Somos, Jun 11 2007 */

Formula

Expansion of phi(q)^3 * (phi(q)^4 + 7 * 16 * q * psi(q^2)^4) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 24 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(1/2) (t / i)^(7/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A003781. - Michael Somos, Aug 27 2013
Convolution of A005875 and A228746. - Michael Somos, Apr 21 2015

A228745 Expansion of (phi(q)^4 + 7 * phi(-q)^4) / 8 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 24, -24, 24, -36, 96, -48, 24, -78, 144, -72, 96, -84, 192, -144, 24, -108, 312, -120, 144, -192, 288, -144, 96, -186, 336, -240, 192, -180, 576, -192, 24, -288, 432, -288, 312, -228, 480, -336, 144, -252, 768, -264, 288, -468, 576, -288, 96, -342, 744
Offset: 0

Views

Author

Michael Somos, Sep 02 2013

Keywords

Comments

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

Examples

			G.f. = 1 - 6*q + 24*q^2 - 24*q^3 + 24*q^4 - 36*q^5 + 96*q^6 - 48*q^7 + 24*q^8 + ...

Links

Crossrefs

Cf. A003781, A004011, A005875, A008438, A228746.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(4), 2), 51); A[1] - 6*A[2]; /* Michael Somos, Aug 21 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 4, 0, q]^4) / 8, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( (A^4 + 7 * subst(A, x, -x)^4) / 8, n))};

Formula

a(n) = -6 * b(n) where b() is multiplicative with a(0) = 1, b(2^e) = -4 if e>1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228746.
G.f.: ( (Sum_{k in Z} x^k^2)^4 + 7 * (Sum_{k in Z} (-x)^k^2)^4 ) / 8.
a(2*n) = A004011(n). a(2*n + 1) = -6 * A008438(n).
Convolution with A005875 is A003781.

A030443 Nonzero coefficients in theta series of {E_7}* lattice.

Original entry on oeis.org

1, 56, 126, 576, 756, 1512, 2072, 4032, 4158, 5544, 7560, 12096, 11592, 13664, 16704, 24192, 24948, 27216, 31878, 44352, 39816, 41832, 55944, 72576, 66584, 67536, 76104, 100800, 99792, 101304, 116928, 145728, 133182, 126504, 160272, 205632, 177660, 176456, 205128, 249984, 249480, 234360
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In the Eichler and Zagier reference this is e_4(A014601(n)), n >= 0, (p. 141), where e_4 is obtained from e_{4,1}(n,r), eq. (7), p. 22, depending only on 4*n-r^2 >= 0 (for integers n and r), i.e. on A014601(n), n >= 0 (with a new notation for n). - Wolfdieter Lang, Jan 08 2016

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.
  • M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhäuser, 1985, p. 141.

Links

Crossrefs

Cf. A003781.

Programs

  • PARI
    f(n) = local(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A^3 * (A^4 + 7 * subst(A, x, -x)^4) / 8, n)); \\ A003781
lista(nn) = select(x->(x>0), vector(nn, k, f(k-1))); \\ Michel Marcus, Nov 11 2023

Extensions

More terms from Michel Marcus, Nov 11 2023
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.