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.

A098098 a(n) = sigma(6*n+5)/6.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 8, 9, 10, 14, 12, 16, 14, 15, 20, 17, 18, 19, 24, 26, 22, 23, 28, 25, 32, 32, 28, 29, 30, 38, 32, 33, 40, 40, 44, 42, 38, 39, 40, 57, 42, 43, 44, 45, 62, 47, 56, 49, 56, 62, 52, 53, 60, 64, 68, 64, 58, 59, 60, 74, 72, 70, 64, 65, 80, 67, 76, 80, 70, 93, 72
Offset: 0

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Author

Vladeta Jovovic, Sep 14 2004

Keywords

Comments

Euler transform of period 6 sequence [2, 0, 0, 0, 2, -4, ...].
Expansion of q^(-5/6) * (eta(q)^-1 * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q. - Michael Somos, Sep 16 2004
2*a(n) is the number of bipartitions of 2*n+1 that are 3-cores. See Baruah and Nath. - Michel Marcus, Apr 13 2020

Examples

			G.f. =1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 + ...
G.f. = q^5 + 2*q^11 + 3*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 7*q^41 + 8*q^47 + 9*q^53 + ...

Links

Crossrefs

Cf. A000203, A016969, A033686, A097723, A121443, A125514, A133739, A134077, A134078, A134079, A186100, A232343, A232356, A252650, A238831.

Programs

  • Magma
    Basis( ModularForms( Gamma0( 36), 2), 432)[6]; /* Michael Somos, Jul 09 2018 */
  • Mathematica
    Table[DivisorSigma[1, 6 n + 5]/6, {n, 0, 71}] (* Ivan Neretin, Apr 30 2016 *)
  • PARI
    a(n) = sigma(6*n + 5)/6
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A))^2, n))} /* Michael Somos, Sep 16 2004 */

Formula

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)))^2. - Michael Somos, Sep 16 2004
From Michael Somos, Jul 09 2018: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252650. -
Convolution square of A121444.
A232343(2*n) = (-1)^n * A258831(n) = A000203(6*n + 4) = a(n). A033686(2*n) = -A134079(2*n + 1) = 2 * a(n). A121443(6*n + 5) = A133739(6*n + 5) = A232356(6*n + 5) = A134077(3*n + 2) = 6 * a(n). A125514(6*n + 5) = 24 * a(n). A134078(6*n + 5) = -36 * a(n). A186100(6*n + 5) = -72 * a(n). (End)
From Amiram Eldar, Dec 16 2022: (Start)
a(n) = A000203(A016969(n))/6.
Sum_{k=1..n} a(k) = (Pi^2/18) * n^2 + O(n*log(n)). (End)

A134078 Expansion of (phi(-q) / phi(-q^2))^3 * phi(q^3)^5 / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 18, -34, 42, -36, 30, -48, 90, -118, 108, -72, 54, -84, 144, -204, 186, -108, 66, -120, 252, -272, 216, -144, 102, -186, 252, -370, 336, -180, 180, -192, 378, -408, 324, -288, 90, -228, 360, -476, 540, -252, 240, -264, 504, -708, 432, -288, 198, -342
Offset: 0

Views

Author

Michael Somos, Oct 06 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 6*x + 18*x^2 - 34*x^3 + 42*x^4 - 36*x^5 + 30*x^6 - 48*x^7 + 90*x^8 + ...

Links

Crossrefs

Cf. A098098, A133739, A134079.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, -q^2])^3*(EllipticTheta[3, 0, q^3]^5/EllipticTheta[3, 0, -q^6]), {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 22 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^4 + A)^3 * eta(x^6 + A)^23 / ( eta(x^2 + A)^9 * eta(x^3 + A)^10 * eta(x^12 + A)^9 ), n))};

Formula

Euler transform of period 12 sequence [ -6, 3, 4, 0, -6, -10, -6, 0, 4, 3, -6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 8 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133739.
a(3*n + 2) = 18 * A134079(n). a(6*n + 5) = -36 * A098098(n).

A329651 Expansion of x * (psi(x^6) / psi(-x^3))^3 * phi(-x)^5 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

0, 1, -9, 31, -45, 6, 45, 8, -117, 121, -54, 12, 9, 14, -72, 186, -261, 18, 207, 20, -270, 248, -108, 24, -63, 31, -126, 391, -360, 30, 270, 32, -549, 372, -162, 48, 171, 38, -180, 434, -702, 42, 360, 44, -540, 726, -216, 48, -207, 57, -279, 558, -630, 54, 693
Offset: 0

Views

Author

Michael Somos, Nov 18 2019

Keywords

Comments

Number 105 of the 126 eta-quotients listed in Table 1 of Williams 2012.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 144 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A328788.

Examples

			G.f. = x - 9*x^2 + 31*x^3 - 45*x^4 + 6*x^5 + 45*x^6 + 8*x^7 - 117*x^8 + ...

Links

Crossrefs

Cf. A000203, A098098, A133739, A134077, A328788.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(12), 2), 52); A[2] - 9*A[3] + 31*A[4] - 45*A[5];
  • Mathematica
    a[ n_] := SeriesCoefficient[ 1/2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, Pi/4, x^(3/2)])^3 EllipticTheta[ 4, 0, x]^5 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}] // PowerExpand;
  • PARI
    {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, 0, s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4))};
  • PARI
    {a(n) = my(A); if( n < 1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^9 * eta(x^12 + A)^3 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^4 + A)), n))};

Formula

Euler transform of period 12 sequence [-9, -5, -6, -4, -9, -2, -9, -4, -6, -5, -9, -4, ...].
Expansion of x * phi(-x)^5 / psi(-x) * (f(-x^12) / f(-x^3))^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q)^9 * eta(q^12)^3 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^4)) in powers of q.
a(n) = s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4) if n>0 where s(x) = sum of divisors of x for integer x else 0.
a(n) = -(-1)^n * A133739(n). a(4*n + 2) = -9 * A134077(n). a(6*n + 5) = 6 * A098098(n).
Showing 1-3 of 3 results.

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