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.

A282937 a(n) = A000728(5*n).

Original entry on oeis.org

1, -6, 9, 10, -30, 1, 5, 51, 10, -100, 20, -55, 109, 110, -130, -1, -110, 160, 10, -230, 100, 15, 191, 120, -230, -100, -89, 160, 90, -340, 120, 5, 300, 200, -260, -1, -275, 240, -100, -270, 119, -165, 260, 410, -200, -40, 20, 200, -110, -500, 180, -54, 140
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2017

Keywords

Comments

This generalized function is related to two following identities; R(q^5) - q - q^2/R(q^5) = (q; q){infinity}/(q^25; q^25){infinity}, R^5(q^5) - 11*q^5 - q^10/R^5(q^5) = ((q^5; q^5){infinity}/(q^25; q^25){infinity})^6, where R(q) is the Rogers-Ramanujan continued function and (q; q)_n is the q-Pochhammer symbol. See the reference.

Examples

			G.f.: 1 - 6*q + 9*q^2 + 10*q^3 - 30*q^4 + q^5 + 5*q^6 + 51*q^7 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 185.

Links

Crossrefs

Cf. A000728, A182821 (Product_{n>=1} (1 - q^(5*n))/(1 - q^n)^6), A282941.
Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), this sequence (k=5), A282942 (k=7).

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^6/(1 - x^(5*j)): j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    CoefficientList[Series[Product[(1 - x^j)^6/(1 - x^(5*j)), {j,1,62}], {x,0,60}], x] (* G. C. Greubel, Nov 18 2018 *)
  • PARI
    m=60; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^j)^6/(1 - x^(5*j)))) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 60
x = R.gen().O(prec)
s = prod((1 - x^j)^6/(1 - x^(5*j)) for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - q^n)^6/(1 - q^(5*n)).
a(n) = (-1)^j mod 5 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 5.
Sum_{k=0..n} a(k)*A182821(n-k) = 0 for n > 0. - Seiichi Manyama, Feb 28 2017
G.f.: exp( Sum_{n>=1} -sigma(5*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = -(1/n)*Sum_{k=1..n} sigma(5*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A282942 Expansion of Product_{k>=1} (1 - q^k)^8/(1 - q^(7*k)) in powers of q.

Original entry on oeis.org

1, -8, 20, 0, -70, 64, 56, 1, -133, -140, 308, -70, 174, 56, -518, -141, -63, 868, -140, 238, 294, -1029, -1154, -203, 2366, -658, 1296, 350, -1547, -1295, -1666, 3234, -2128, 2534, 2464, -2577, -3087, -609, 5600, -2716, 2435, 294, -3745, -4249, -1015, 8526
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2017

Keywords

Examples

			G.f.: 1 - 8*q + 20*q^2 - 70*q^4 + 64*q^5 + 56*q^6 + q^7 - 133*q^8 - 140*q^9 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 192.

Links

Crossrefs

Cf. A282941.
Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), A282937 (k=5), this sequence (k=7).

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^j)^8/(1-x^(7*j)): j in [1..m]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^k)^8/(1-x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Nov 18 2018 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(j=1,m, (1-x^j)^8/(1-x^(7*j)))) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = prod((1-x^j)^8/(1-x^(7*j)) for j in (1..50))
list(s) # G. C. Greubel, Nov 18 2018

Formula

G.f.: exp( Sum_{n>=1} -sigma(7*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = -(1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A116916 Expansion of q^(-1/8) * (eta(q)^3 + 3 * eta(q^9)^3) in powers of q^3.

Original entry on oeis.org

1, 5, -7, 0, 0, -11, 0, 13, 0, 0, 0, 0, 17, 0, 0, -19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Feb 26 2006

Keywords

Comments

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

Examples

			1 + 5*x - 7*x^2 - 11*x^5 + 13*x^7 + 17*x^12 - 19*x^15 - 23*x^22 + 25*x^26 + ...
q + 5*q^25 - 7*q^49 - 11*q^121 + 13*q^169 + 17*q^289 - 19*q^361 +...

Links

Crossrefs

Cf. A010816, A202394.
Cf. A282937, A282941.

Programs

  • Mathematica
    a[0] = 1; a[n_] := SeriesCoefficient[QPochhammer[x + x*O[x]^(3n)]^3 + 3x * QPochhammer[x^9 + O[x]^(3n)]^3, 3n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 06 2015, adapted from PARI *)
a[ n_] := With[ {m = Sqrt[ 24 n + 1]}, If[ IntegerQ[ m], m KroneckerSymbol[ 3, m] KroneckerSymbol[ -3, m], 0]]; (* Michael Somos, Apr 27 2018 *)
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( 3, n) * kronecker( -3, n))};
  • PARI
    {a(n) = if( n<1, n==0, n*=3; polcoeff( eta(x + x * O(x^n))^3 + 3 * x * eta(x^9 + x * O(x^n))^3, n))};

Formula

Expansion of f(-x) * a(x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
Expansion of f(-x)^3 + 3 * x * f(-x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 4608^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A202394.
G.f.: Sum_{k in Z} (-1)^k * (6*k + 1) * x^(k * (3*k + 1) / 2).
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = 5 * a(n).
a(n) = A010816(3*n).
Showing 1-3 of 3 results.

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