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 1-2 of 2 results.

A208851 Partitions of 2*n + 1 into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

Original entry on oeis.org

1, 3, 6, 11, 20, 34, 56, 91, 143, 220, 334, 498, 732, 1064, 1528, 2171, 3058, 4269, 5910, 8124, 11088, 15034, 20264, 27154, 36189, 47988, 63324, 83176, 108780, 141672, 183776, 237499, 305812, 392406, 501856, 639781, 813108, 1030354, 1301928, 1640572, 2061850
Offset: 0

Views

Author

Michael Somos, Mar 02 2012

Keywords

Comments

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

Examples

			1 + 3*q + 6*q^2 + 11*q^3 + 20*q^4 + 34*q^5 + 56*q^6 + 91*q^7 + 143*q^8 + ...
a(2) = 6 since  2*2 + 1 = 5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.

Links

Crossrefs

Cf. A185083, A208850.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q^2]/EllipticTheta[3, 0, -q] - 1)/(2*q), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 05 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, n = 2*n + 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}

Formula

Expansion of (phi(q^2) / phi(-q) - 1) / (2 * q) in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 16 sequence [ 3, 0, 1, 2, 1, 2, 3, 0, 3, 2, 1, 2, 1, 0, 3, 0, ...].
2 * a(n) = A208850(n + 1). a(n) = A185083(n + 1).

A214639 Expansion of q * f(-q^2, -q^14) / f(-q, q^3) in powers of q where f(,) is Ramanujan's two-variable theta function.

Original entry on oeis.org

0, 1, 1, 0, -1, -2, -2, 0, 3, 5, 4, 0, -6, -10, -8, 0, 11, 18, 15, 0, -20, -32, -26, 0, 34, 55, 44, 0, -56, -90, -72, 0, 91, 144, 114, 0, -143, -226, -178, 0, 220, 346, 272, 0, -334, -522, -408, 0, 498, 777, 605, 0, -732, -1138, -884, 0, 1064, 1648, 1276, 0
Offset: 0

Views

Author

Michael Somos, Jul 24 2012

Keywords

Comments

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

Examples

			q + q^2 - q^4 - 2*q^5 - 2*q^6 + 3*q^8 + 5*q^9 + 4*q^10 - 6*q^12 - 10*q^13 + ...

Links

Crossrefs

Cf. A079006, A185083, A210063, A214263, A224216.

Programs

  • Mathematica
    nn = 16*10; b = Flatten[Table[{1, -1, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, -1, 1, 0}, {nn/16}]]; CoefficientList[x * Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jul 25 2012 *)
a[ n_] :=  SeriesCoefficient[ 2 q^(9/8) QPochhammer[ -q, q^8]^2 QPochhammer[ q^8]^2 QPochhammer[ -q^7, q^8]^2 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q^4]), {q, 0, n}] (* Michael Somos, Apr 02 2013 *)
  • PARI
    {a(n) = local(A); if( n<1, 0, if( n%4==3, 0, if( n%2==0, n/=2; A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(-x + A) / eta(x + A), n) / 2, n\=4; A = x * O(x^n); (-1)^n * polcoeff( (eta(x^4 + A) / eta(x + A))^2, n))))}
  • PARI
    {a(n) = if( n<1, 0, n--; polcoeff( prod( k=1, n, (1 -x^k + x * O(x^n))^[ 0, -1, 1, 1, 1, 1, -1, -1, -2, -1, -1, 1, 1, 1, 1, -1][k%16 + 1]), n))}

Formula

Expansion of q * f(q, q^7)^2 / (phi(q^4) * psi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 16 sequence [ 1, -1, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, -1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 2 * u * v * (1 - u - u*v).
a(4*n + 3) = 0. a(4*n) = (-1)^n * A185083(n) unless n=0. a(4*n + 1) = A079006(n). a(4*n + 2) = A210063(n).
a(2*n) = A224216(n).
Showing 1-2 of 2 results.

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