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.

A224822 Expansion of phi(-q) * phi(-q^3)^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, -4, 10, 0, 4, -16, 0, -2, 8, 0, 12, -8, 0, -16, 26, 0, 0, -24, 0, -8, 8, 0, 20, -10, 0, -4, 32, 0, 8, -48, 0, -8, 16, 0, 10, -8, 0, -32, 40, 0, 8, -24, 0, 0, 16, 0, 28, -18, 0, -24, 40, 0, 4, -64, 0, -8, 8, 0, 32, -24, 0, -16, 58, 0, 16, -24, 0, -16
Offset: 0

Views

Author

Michael Somos, Jul 20 2013

Keywords

Comments

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

Examples

			G.f. = 1 - 2*q - 4*q^3 + 10*q^4 + 4*q^6 - 16*q^7 - 2*q^9 + 8*q^10 + 12*q^12 +
...

Links

Crossrefs

Cf. A028967, A224821.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^4 / (eta(x^2 + A) * eta(x^6 + A)^2), n))};

Formula

Expansion of eta(q)^2 * eta(q^3)^4 / (eta(q^2) * eta(q^6)^2) in powers of q.
Euler transform of period 6 sequence [-2, -1, -6, -1, -2, -3, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^k^2) * (Sum_{k in Z} (-1)^k * x^(3*k^2))^2.
a(3*n + 2) = 0. a(2*n) = A028967(n). a(3*n) = A224821(n).

A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

Original entry on oeis.org

1, -1, -2, -2, 3, 8, 0, -2, -10, -4, 2, 4, 10, -8, -4, 0, 7, 12, 4, -2, -16, -16, 4, 8, 0, -7, -4, -2, 10, 24, 8, -2, -26, 0, 2, 8, 12, -16, -8, -8, 10, 12, 0, -6, -20, -16, 4, 8, 26, -7, -10, 0, 16, 40, 0, -4, -20, -24, 6, 4, 0, -16, -12, -8, 15, 24, 8, -6
Offset: 0

Views

Author

Michael Somos, Nov 26 2019

Keywords

Examples

			G.f. = 1 - x - 2*x^2 - 2*x^3 + 3*x^4 + 8*x^5 - 2*x^7 - 10*x^8 - 4*x^9 + ...

Crossrefs

Cf. A028967, A030206, A195848, A224822, A329956, A329957.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3]^3 / QPochhammer[ x^6]^2, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^3 / eta(x^6 + A)^2, n))};

Formula

Euler transform of period 6 sequence [-1, -2, -4, -2, -1, -3, ...].
G.f.: Product_{k>=1} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 + x^(3*k))^2.
Convolution of A030206 and A195848.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 1990656^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329958.
a(3*n) = A224822(n). a(3*n + 1) = -A329956(n). a(3*n + 2) = -2*A329957(n). a(6*n) = A028967(n).

A028966 Norms of vectors in the a.c.c. lattice, divided by 2.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 83, 84, 86, 87, 89, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 110, 111, 113
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Equivalently, numbers represented by quadratic form with Gram matrix [ 4, 2, 1; 2, 4, 2; 1, 2, 4 ], divided by 2.
See A028967 for further information.

Examples

			1 + 10*q^4 + 4*q^6 + 8*q^10 + 12*q^12 + 26*q^16 + 8*q^22 + 20*q^24 + 32*q^28 + 8*q^30 + ...

Links

Programs

  • Magma
    L:=LatticeWithGram(3, [4,-1,-1, -1,4,-2, -1,-2,4]); T := ThetaSeries(L,500); T;
  • Maple
    L := [seq(0,i=1..1)]: for x from -20 to 20 do for y from -20 to 20 do for z from -20 to 20 do if member(4*x^2+4*x*y+2*x*z+4*y^2+4*y*z+4*z^2, L)=false then L := [op(L), 4*x^2 +4*x*y+2*x*z+4*y^2+4*y*z+4*z^2] fi: od: od: od: L2 := sort(L): for i from 1 to 100 do printf(`%d, `,L2[i]/2) od: # James Sellers, May 31 2000

Extensions

More terms from James Sellers, May 31 2000
Edited by N. J. A. Sloane, Sep 29 2006
Showing 1-3 of 3 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.