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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A008425 Theta series of {D_6}* lattice.

Original entry on oeis.org

1, 0, 12, 64, 60, 0, 160, 384, 252, 0, 312, 960, 544, 0, 960, 1664, 1020, 0, 876, 2880, 1560, 0, 2400, 4224, 2080, 0, 2040, 5248, 3264, 0, 4160, 7680, 4092, 0, 3480, 9984, 4380, 0, 7200, 10880, 6552, 0, 4608, 14784, 8160, 0, 10560, 17664, 8224, 0, 7812, 18560
Offset: 0

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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 + 12*x^2 + 64*x^3 + 60*x^4 + 160*x^6 + 384*x^7 + 252*x^8 + 312*x^10 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag New York, 1999, ISBN 978-1-4757-6568-7, p. 120.

Links

Crossrefs

Cf. A000141, A008428.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(8), 3), 52); A[1] + 12*A[3] + 64*A[4] + 60*A[5] + 160*A[7]; /* Michael Somos, Dec 14 2016 */
  • Mathematica
    a[n_] := DivisorSum[n, #^2*(4*(KroneckerSymbol[-4, n/#]-KroneckerSymbol[-4, #]))&]; a[0]=1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 06 2016, after Ralf Stephan *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2]^6 + EllipticTheta[ 2, 0, x^2]^6, {x, 0, n}]; (* Michael Somos, Dec 14 2016 *)
  • PARI
    {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (kronecker(-4, n/d) - kronecker(-4, d))))}; /* Michael Somos, Dec 14 2016 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / (eta(x^2 + A) * eta(x^8 + A))^2)^6 + 64 * x^3 * (eta(x^8 + A)^2/ eta(x^4 + A))^6, n))}; /* Michael Somos, Dec 14 2016 */
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [2, 0, 0, 0, 0, 1; 0, 2, 0, 0, 0, 1; 0, 0, 2, 0, 0, 1; 0, 0, 0, 2, 0, 1; 0, 0, 0, 0, 2, 1; 1, 1, 1, 1, 1, 3]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Dec 14 2016 */

Formula

Apparently, a(n) = Sum_{d|n} d^2*(4*(Kronecker(-4,n/d) - Kronecker(-4,d))), n > 0. - Ralf Stephan, Dec 31 2014
Expansion of phi(x^2)^6 + 64 * x^3 * psi(x^4)^6 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Dec 14 2016
G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 16 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008428. - Michael Somos, Dec 14 2016
G.f.: theta_3(0, x^2)^6 + theta_2(0, x^2)^6.

A297331 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 4, 2, 0, 1, 12, 4, 0, 0, 1, 24, 6, 0, 0, 0, 1, 40, 24, 24, 4, 0, 0, 1, 60, 90, 96, 12, 8, 0, 0, 1, 84, 252, 240, 24, 24, 0, 0, 0, 1, 112, 574, 544, 200, 144, 8, 0, 2, 0, 1, 144, 1136, 1288, 1020, 560, 96, 48, 4, 0, 0, 1, 180, 2034, 3136, 3444, 1560, 400, 192, 6, 4, 0, 0
Offset: 0

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Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Examples

			Square array begins:
1,  1,  1,   1,    1,    1,  ...
0,  0,  4,  12,   24,   40,  ...
0,  2,  4,   6,   24,   90,  ...
0,  0,  0,  24,   96,  240,  ...
0,  0,  4,  12,   24,  200,  ...
0,  0,  8,  24,  144,  560,  ...

Links

Crossrefs

Columns k=0..32 give A000007, A089798 (absolute values), A004018, A004015, A004011, A005930, A008428, A008429, A008430, A008431, A008432, A022042, A022043, A022044, A022045, A022046, A022047, A022048, A022049, A022050, A022051, A022052, A022053, A022054, A022055, A022056, A022057, A022058, A022059, A022060, A022061, A022062, A022063.
Main diagonal gives A303333.
Cf. A122141, A286815.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[(EllipticTheta[3, 0, q^(1/2)]^k + EllipticTheta[4, 0, q^(1/2)]^k)/2, {q, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

G.f. of column k: (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2, where theta_() is the Jacobi theta function.
Showing 1-2 of 2 results.

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

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