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.

Previous Showing 11-13 of 13 results.

A286953 Expansion of Product_{j>=1} (1 - x^j)/(1 - x^(4*j))^4.

Original entry on oeis.org

1, -1, -1, 0, 4, -3, -4, 1, 14, -10, -14, 4, 39, -26, -40, 13, 101, -65, -105, 36, 238, -147, -251, 91, 534, -322, -569, 212, 1135, -666, -1222, 469, 2328, -1340, -2526, 987, 4606, -2600, -5035, 2002, 8867, -4928, -9751, 3926, 16624, -9100, -18382, 7488, 30499
Offset: 0

Views

Author

Seiichi Manyama, May 17 2017

Keywords

Links

Crossrefs

4th column of A286950.
Cf. A045831.

A153182 Number of overpartitions of n with even M2-rank minus the number of overpartitions of n with odd M2-rank.

Original entry on oeis.org

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

Views

Author

Jeremy Lovejoy, Dec 20 2008

Keywords

Comments

a(8n+3) = 0.

Examples

			G.f. = 1 + 2*x + 4*x^2 - 2*x^4 + 8*x^5 + 8*x^6 - 8*x^7 - 4*x^8 + ...

Links

Crossrefs

Cf. A005875, A045831, A213022, A213624, A213625, A256624.

Programs

  • Mathematica
    a[n_]:= If[n < 0, 0, SeriesCoefficient[(1 + 8*Sum[(-1)^k x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, (Sqrt[4 n + 1] - 1)/2}])/EllipticTheta[4, 0, x], {x, 0, n}]]; Table[a[n], {n,0,30}] (* G. C. Greubel, Nov 29 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + 8 * sum(k=1, sqrtint(n+1)-1, (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2, A)) / (1 + 2 * sum(k=1, sqrtint(n), (-1)^k * x^k^2, A)), n))}; /* Michael Somos, Jul 13 2015 */
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, n, x^k / (1 + x^(2*k)) * prod(j=1, k,  (1 + x^(2*j - 1)) / (1 + x^(2*j)), 1 + x * O(x^(n-k)))), n))}; /* Michael Somos, Jul 13 2015 */

Formula

G.f.: 1 + 2Sum_{n >= 1} q^n(1+q)(1+q^3)...(1+q^(2n-1))/((1+q^2)(1+q^4)...(1+q^(2n))^2).
G.f.: (1 + 8 * Sum_{k>0} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (1 + 2 * Sum_{k>0} (-1)^k * x^k^2). - Michael Somos, Jul 13 2015
a(4*n) = A256624(n) = 2 * a(n) - A005875(n). - Michael Somos, Jul 13 2015
a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = 4 * A213625(n). a(8*n + 5) = 8 * A045831(n). a(8*n + 6) = 8 * A213624(n). - Michael Somos, Jul 13 2015

A246631 Number of integer solutions to x^2 + 2*y^2 + 2*z^2 = n.

Original entry on oeis.org

1, 2, 4, 8, 6, 8, 8, 0, 12, 10, 8, 24, 8, 8, 16, 0, 6, 16, 12, 24, 24, 16, 8, 0, 24, 10, 24, 32, 0, 24, 16, 0, 12, 16, 16, 48, 30, 8, 24, 0, 24, 32, 16, 24, 24, 24, 16, 0, 8, 18, 28, 48, 24, 24, 32, 0, 48, 16, 8, 72, 0, 24, 32, 0, 6, 32, 32, 24, 48, 32, 16, 0
Offset: 0

Views

Author

Michael Somos, Aug 31 2014

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^2 + 8*q^3 + 6*q^4 + 8*q^5 + 8*q^6 + 12*q^8 + 10*q^9 + ...

Links

Crossrefs

Cf. A004015, A005875, A008443, A014455, A033717, A045826, A045828, A045831, A045834, A212885, A213022, A213624, A213625, A246811.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(8), 3/2), 80); A[1] + 2*A[2];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^2, {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 2, 0; 0, 0, 2], n)[n])};
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^8 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};

Formula

Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 2, 0; 0, 0, 2 ].
Expansion of phi(q) * phi(q^2)^2 = phi(-q^4)^4 / phi(-q) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q^2) * eta(q^4)^8 / (eta(q)^2 * eta(q^8)^4) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, -7, 2, 1, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 4 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A014455.
G.f.: theta_3(q) * theta_3(q^2)^2.
G.f.: Product{k>0} (1 - x^(2*k)) * (1 - x^(4*k))^8 / ((1 - x^k)^2 * (1 - x^(8*k))^4).
G.f.: Product{k>0} (1 + x^(2*k)) * (1 + x^k)^2 * (1 - x^(4*k))^3 / (1 + x^(4*k))^4.
a(n) = (-1)^floor((n+1) / 2) * A212885(n) = abs(A212885(n)).
a(n) = A033717(2*n). a(2*n) = A014455(n). a(2*n + 1) = 2 * A246811(n).
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = 4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = 4 * A213625(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). a(8*n + 5) = 8 * A045831(n). a(8*n + 6) = 8 * A213624(n). a(8*n + 7) = 0.
Previous Showing 11-13 of 13 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.