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.

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A029751 Average theta series of odd unimodular lattices in dimension 12.

Original entry on oeis.org

1, 8, 248, 1952, 7928, 25008, 60512, 134464, 253688, 474344, 775248, 1288416, 1934432, 2970352, 4168384, 6101952, 8118008, 11358864, 14704664, 19808800, 24782928, 32809216, 39940896, 51490752, 61899872, 78150008, 92080912
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • R. A. Rankin, Modular Forms, p. 240 ff.
  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.

Links

Crossrefs

Cf. A000145, A000735.

Programs

  • Mathematica
    a[0] = 1; a[n_] := (-1)^(n-1)*8*DivisorSum[n, (-1)^(n + n/#)*#^5&]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 06 2017, translated from PARI *)
  • PARI
    a(n)=if(n<1, n==0, (-1)^(n-1)*8*sumdiv(n,d,(-1)^(n+n/d)*d^5)) /* Michael Somos, Sep 21 2005 */

Formula

G.f.: 1 + 8*Sum_{k>0} k^5 x^k/(1+(-x)^k). - Michael Somos, Sep 21 2005
A000145(n) = a(n) + 16*A000735(n). - Michael Somos, Sep 21 2005

A209676 Expansion of f(x)^12 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 12, 54, 88, -99, -540, -418, 648, 594, -836, 1056, 4104, -209, -4104, -594, -4256, -6480, 4752, -298, -5016, 17226, 12100, -5346, 1296, -9063, 7128, 19494, -29160, -10032, 7668, -34738, -8712, -22572, -21812, 49248, 46872, 67562, -2508, -47520, 76912
Offset: 0

Views

Author

Michael Somos, Mar 11 2012

Keywords

Comments

Number 35 of the 74 eta-quotients listed in Table I of Martin (1996). See g.f. B(q) below: cusp form weight 6 level 16.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 12*x + 54*x^2 + 88*x^3 - 99*x^4 - 540*x^5 - 418*x^6 + 648*x^7 + ...
G.f. B(q) of {b(n)}: q + 12*q^3 + 54*q^5 + 88*q^7 - 99*q^9 - 540*q^11 - 418*q^13 + ...

Links

Crossrefs

Cf. A121373, A133089, A187076.
A000735 is the same except for signs.

Programs

  • Magma
    A := Basis( CuspForms( Gamma0(16), 6), 81); A[1] + 12*A[3] + 54*A[5] + 88*A[7]; /* Michael Somos, Jun 09 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^12, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^12, n))};

Formula

Expansion of q^(-1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^12 in powers of q.
Euler transform of period 4 sequence [ 12, -24, 12, -12, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4096 (t/i)^6 f(t) where q = exp(2 Pi i t).
G.f.: (Product_{k>0} (1 - (-x)^k))^12.
a(n) = (-1)^n * A000735(n).
Convolution cube of A187076. Convolution fourth power of A133089. Convolution twelfth power of A121373.

A225923 Expansion of q^(-1/2) * k(q) * (1 - k(q)^4) * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions.

Original entry on oeis.org

1, 20, -74, -24, 157, 124, 478, -1480, -1198, 3044, -480, 184, 2351, -1720, -3282, -5728, 2480, 1776, 10326, 9560, -8886, -9188, -11618, 23664, -16231, -23960, 11686, -9176, 60880, 16876, -18482, -3768, -35372, -15532, 3680, -31960, -4886, 47020, -2976, 44560
Offset: 0

Views

Author

Michael Somos, May 20 2013

Keywords

Comments

In Glaisher (1907) denoted by gamma(m) defined in section 63 on page 38.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In Chan and Combes (2024) page 2 is g(z) = eta(z)^8 eta(4z)^4 + 8 eta(4z)^12 identified as the unique newform of weight 6 and level 8 with LMFDB label 8.6.a.a. - Michael Somos, Jun 25 2025

Examples

			G.f. = 1 + 20*x - 74*x^2 - 24*x^3 + 157*x^4 + 124*x^5 + 478*x^6 - 1480*x^7 + ...
G.f. = q + 20*q^3 - 74*q^5 - 24*q^7 + 157*q^9 + 124*q^11 + 478*q^13 - 1480*q^15 + ...

References

  • J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 38).

Links

Crossrefs

Cf. A000735, A002292, A225872.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q]^12 + 32 q (QPochhammer[ q] QPochhammer[ q^4]^2)^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 8 q QPochhammer[ q^4]^12 + (QPochhammer[ q]^2 QPochhammer[ q^4])^4, {q, 0, 2 n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^12 + 32 * x * eta(x + A)^4 * eta(x^4 + A)^8, n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( 8 * x * eta(x^4 + A)^12 + eta(x + A)^8 * eta(x^4 + A)^4, n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 / (eta(x + A) * eta(x^4 + A)^2))^4 + 16 * x * (eta(x + A) * eta(x^4 + A)^2)^4, n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2  + A)^12 / (eta(x + A)^5 * eta(x^4 + A)^4))^4 - x^2 * (4 * eta(x^4 + A)^4 / eta(x + A))^4, n))}; /* Michael Somos, Jul 20 2013 */

Formula

Expansion of (psi(x) * phi(-x^2)^2)^4 + 16 * x * (psi(x) * psi(-x)^2)^4 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (phi(x)^8 - 256 * x^2 * psi(x^2)^8) * psi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jul 20 2013
Expansion of q^(-1/2) * (eta(q)^12 + 32 * q * eta(q)^4 * eta(q^4)^8) in powers of q.
Expansion of q^(-1) * eta(q^4)^4 * (eta(q)^8 + 8 * eta(q^4)^8) in power of q^2. - Michael Somos, Jun 25 2025
G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = 512 * (t/i)^6 * 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(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) if p > 2.
G.f.: Product_{k>0} (1 - x^k)^12 + 32 * x * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2)^4.
|a(n)| = A002292(n). a(n) = A000735(n) + 32 * A225872(n).

A227239 Expansion of q * f(-q^2)^12 + 8 * q^2 * f(-q^4)^12 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 8, -12, 0, 54, -96, -88, 0, -99, 432, 540, 0, -418, -704, -648, 0, 594, -792, 836, 0, 1056, 4320, -4104, 0, -209, -3344, 4104, 0, -594, -5184, 4256, 0, -6480, 4752, -4752, 0, -298, 6688, 5016, 0, 17226, 8448, -12100, 0, -5346, -32832, -1296, 0, -9063
Offset: 1

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).
The bisection {a(2*n+1)} is Glaisher's Omega function, A000735. The other bisection, {a(2*n)}, begins 8, 0, -96, 0, 432, 0, -704, 0, -792, 0, 4320, 0, -3344, ..., and if this in turn is bisected and then divided by 8, we again obtain A000735. - N. J. A. Sloane, Nov 25 2018

Examples

			q + 8*q^2 - 12*q^3 + 54*q^5 - 96*q^6 - 88*q^7 - 99*q^9 + 432*q^10 + ...

Links

Crossrefs

Cf. A000735.

Programs

  • Magma
    A := Basis( CuspidalSubspace( ModularForms( Gamma1(8), 6))); PowerSeries( A[1] +8*A[2] -12*A[3] +54*A[5] -96*A[6] -88*A[7], 50);
  • Mathematica
    a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^12 + 8 q^2 QPochhammer[ q^4]^12, {q, 0, n}]
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^12 + 8 * x * eta(x^4 + A)^12, n))}

Formula

Expansion of eta(q^2)^12 + 8 * eta(q^4)^12 in powers of q.
a(n) is multiplicative with a(2) = 8, a(2^e) = 0 if e > 1, a(p^e) = a(p) * a(p^(e-1)) - p^5 * a(p^(e-2)) if p > 2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^3 (t / i)^6 f(t) where q = exp(2 Pi i t).
a(4*n) = 0. a(2*n + 1) = A000735(n). a(4*n + 2) = 8 * A000735(n).
Previous Showing 11-14 of 14 results.

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

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