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.

A282930 Expansion of Product_{k>=1} (1 - x^(7*k))^48/(1 - x^k)^49 in powers of x.

Original entry on oeis.org

1, 49, 1274, 23275, 334425, 4015011, 41818315, 387605443, 3256150548, 25135003348, 180196297050, 1210028211210, 7663549175191, 46039891115155, 263630633610437, 1444741006154614, 7604013727493190, 38554851707435000, 188824087108333495, 895363849845490543, 4119191297378031000, 18420594133878904635, 80204828814019528689
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^48/(1 - x^j)^49: j in [1..30]]) )); // G. C. Greubel, Nov 18 2018
    
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^48/(1 - x^k)^49, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(prod(j=1,N, (1 - x^(7*j))^48/(1 - x^j)^49)) \\ G. C. Greubel, Nov 18 2018
    
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
    prec = 30
    x = R.gen().O(prec)
    s = prod((1 - x^(7*j))^48/(1 - x^j)^49 for j in (1..prec))
    print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^48/(1 - x^n)^49.
a(n) ~ exp(Pi*sqrt(590*n/21)) * sqrt(295) / (4*sqrt(3) * 7^(49/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282931 Expansion of Product_{k>=1} (1 - x^(7*k))^52/(1 - x^k)^53 in powers of x.

Original entry on oeis.org

1, 53, 1484, 29097, 447426, 5734918, 63638001, 627260142, 5594403499, 45779730871, 347453597091, 2466970932027, 16501339314082, 104588498225862, 631215364345159, 3642533720923593, 20170341090888205, 107511123136305075, 553099301324196585
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^52/(1 - x^j)^53: j in [1..30]]) )); // G. C. Greubel, Nov 18 2018
    
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^52/(1 - x^k)^53, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
  • PARI
    my(N=30,x='x+O('x^N)); Vec(prod(j=1,N, (1 - x^(7*j))^52/(1 - x^j)^53)) \\ G. C. Greubel, Nov 18 2018
    
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
    prec = 30
    x = R.gen().O(prec)
    s = prod((1 - x^(7*j))^52/(1 - x^j)^53 for j in (1..prec))
    print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^52/(1 - x^n)^53.
a(n) ~ exp(Pi*sqrt(638*n/21)) * sqrt(319) / (4*sqrt(3) * 7^(53/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282932 Expansion of Product_{k>=1} (1 - x^(7*k))^56/(1 - x^k)^57 in powers of x.

Original entry on oeis.org

1, 57, 1710, 35815, 586815, 7997157, 94175267, 983458849, 9279004863, 80218101555, 642408637594, 4807304399931, 33855173217278, 225702273908048, 1431470152072364, 8673471170235715, 50389686887219910, 281575909008910196, 1517580284619183809
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Comments

In general, if m >= 1 and g.f. = Product_{k>=1} (1 - x^(7*k))^m / (1 - x^k)^(m+1), then a(n) ~ exp(Pi*sqrt((2*(6*m+7)*n)/21)) * sqrt(6*m+7) / (4*sqrt(3) * 7^((m+1)/2) * n). - Vaclav Kotesovec, Nov 10 2017

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^56/(1 - x^j)^57: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
    
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^56/(1 - x^k)^57, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
  • PARI
    my(N=30,x='x+O('x^N)); Vec(prod(j=1,N, (1 - x^(7*j))^56/(1 - x^j)^57)) \\ G. C. Greubel, Nov 18 2018
    
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
    prec = 30
    x = R.gen().O(prec)
    s = prod((1 - x^(7*j))^56/(1 - x^j)^57 for j in (1..prec))
    print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^56/(1 - x^n)^57.
a(n) ~ exp(Pi*sqrt(686*n/21)) * sqrt(343) / (4*sqrt(3) * 7^(57/2) * n). - Vaclav Kotesovec, Nov 10 2017
Previous Showing 11-13 of 13 results.