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 1-10 of 13 results. Next

A282920 Expansion of Product_{k>=1} (1 - x^(7*k))^8/(1 - x^k)^9 in powers of x.

Original entry on oeis.org

1, 9, 54, 255, 1035, 3753, 12483, 38701, 113193, 315013, 839802, 2155905, 5352252, 12894426, 30233558, 69160869, 154677325, 338822547, 728084435, 1536931932, 3190959918, 6523084815, 13142291319, 26118847655, 51244059231, 99322878506, 190306301025
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^8/(1 - x^j)^9: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^8 /(1 - x^k)^9, {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))^8/(1 - x^j)^9)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^8/(1 - x^j)^9 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^8/(1 - x^n)^9.
a(n) ~ exp(Pi*sqrt(110*n/21)) * sqrt(55) / (4*sqrt(3) * 7^(9/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282921 Expansion of Product_{k>=1} (1 - x^(7*k))^12/(1 - x^k)^13 in powers of x.

Original entry on oeis.org

1, 13, 104, 637, 3276, 14820, 60697, 229360, 810498, 2705118, 8592857, 26134654, 76476816, 216174700, 592220696, 1576826355, 4090222409, 10357895639, 25653139694, 62235901689, 148108568986, 346176981673, 795569268689, 1799508071426, 4009753651904, 8808973137510
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^12/(1 - x^j)^13: j in [1..30]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^12/(1 - x^k)^13, {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))^12/(1 - x^j)^13)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    m = 30
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(m)
s = prod((1 - x^(7*j))^12/(1 - x^j)^13 for j in (1..m))
list(s) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^12/(1 - x^n)^13.
a(n) ~ exp(Pi*sqrt(158*n/21)) * sqrt(79) / (4*sqrt(3) * 7^(13/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282922 Expansion of Product_{n>=1} (1 - x^(7*n))^16/(1 - x^n)^17 in powers of x.

Original entry on oeis.org

1, 17, 170, 1275, 7905, 42619, 206091, 912459, 3753328, 14500320, 53053498, 185046190, 618555931, 1990227519, 6186291009, 18633598578, 54530992072, 155401842842, 432109571275, 1174385295541, 3124445373406, 8148428799893, 20856618453595, 52451748129498
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^16/(1 - x^j)^17: j in [1..30]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^16/(1 - x^k)^17, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
  • PARI
    my(x='x+O('x^30)); Vec(prod(j=1, 30, (1 - x^(7*j))^16/(1 - x^j)^17)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    m = 30
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(m)
s = prod((1 - x^(7*j))^16/(1 - x^j)^17 for j in (1..m))
s.coefficients() # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^16/(1 - x^n)^17.
a(n) ~ exp(Pi*sqrt(206*n/21)) * sqrt(103) / (4*sqrt(3) * 7^(17/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282923 Expansion of Product_{n>=1} (1 - x^(7*n))^20/(1 - x^n)^21 in powers of x.

Original entry on oeis.org

1, 21, 252, 2233, 16170, 100926, 560945, 2837398, 13265679, 57989435, 239125579, 936702879, 3505361650, 12590400326, 43572202835, 145770820937, 472764167939, 1490002933265, 4573182416677, 13694526423445, 40076281579264, 114782535792335, 322167257486123, 887188897987819, 2399619923361150, 6380874322337452, 16695968482412345
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^20/(1 - x^j)^21: j in [1..30]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^20/(1 - x^k)^21, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
  • PARI
    my(x='x+O('x^30)); Vec(prod(j=1, 30, (1 - x^(7*j))^20/(1 - x^j)^21)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^20/(1 - x^j)^21 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^20/(1 - x^n)^21.
a(n) ~ exp(Pi*sqrt(254*n/21)) * sqrt(127) / (4*sqrt(3) * 7^(21/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282924 Expansion of Product_{k>=1} (1 - x^(7*k))^24/(1 - x^k)^25 in powers of x.

Original entry on oeis.org

1, 25, 350, 3575, 29575, 209405, 1312675, 7452201, 38939275, 189537775, 867436570, 3760131375, 15529994130, 61413915500, 233488417752, 856388420815, 3039281123900, 10463551169370, 35024068485525, 114205431037285, 363408170015065, 1130218949978428, 3440267279234290, 10261830946893750, 30029624283800440, 86300123835692431
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^24/(1 - x^j)^25: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^24/(1 - x^k)^25, {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))^24/(1 - x^j)^25)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^24/(1 - x^j)^25 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^24/(1 - x^n)^25.
a(n) ~ exp(Pi*sqrt(302*n/21)) * sqrt(151) / (4*sqrt(3) * 7^(25/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282925 Expansion of Product_{k>=1} (1 - x^(7*k))^28/(1 - x^k)^29 in powers of x.

Original entry on oeis.org

1, 29, 464, 5365, 49880, 394632, 2750969, 17296732, 99742368, 534126988, 2681856693, 12722233068, 57373155952, 247218913828, 1022189562610, 4070289420139, 15656921120982, 58336024110584, 211023516790156, 742643172981206, 2547265600634862, 8529351700138885
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^28/(1 - x^j)^29: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^28/(1 - x^k)^29, {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))^28/(1 - x^j)^29)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^28/(1 - x^j)^29 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^28/(1 - x^n)^29.
a(n) ~ exp(Pi*sqrt(350*n/21)) * sqrt(175) / (4*sqrt(3) * 7^(29/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282926 Expansion of Product_{k>=1} (1 - x^(7*k))^32/(1 - x^k)^33 in powers of x.

Original entry on oeis.org

1, 33, 594, 7667, 79101, 691119, 5299019, 36518791, 230122266, 1343028082, 7331536586, 37731144564, 184232285897, 857974579385, 3827695162667, 16420097827188, 67948512704413, 271990545250303, 1055719283332541, 3981884465793740, 14621550982740229
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^32/(1 - x^j)^33: j in [1..m]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^32/(1 - x^k)^33, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
  • PARI
    my(m=30, x='x+O('x^m)); Vec(prod(j=1,m, (1 - x^(7*j))^32/(1 - x^j)^33)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^32/(1 - x^j)^33 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^32/(1 - x^n)^33.
a(n) ~ exp(Pi*sqrt(398*n/21)) * sqrt(199) / (4*sqrt(3) * 7^(33/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282927 Expansion of Product_{k>=1} (1 - x^(7*k))^36/(1 - x^k)^37 in powers of x.

Original entry on oeis.org

1, 37, 740, 10545, 119510, 1142338, 9548849, 71529474, 488650453, 3084466705, 18173253703, 100751920597, 529029597362, 2645187324766, 12651654794629, 58105915432081, 257102694583806, 1099122519498352, 4551159872375703, 18293134887547452
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^36/(1 - x^j)^37: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^36/(1 - x^k)^37, {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))^36/(1 - x^j)^37)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^36/(1 - x^j)^37 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^36/(1 - x^n)^37.
a(n) ~ exp(Pi*sqrt(446*n/21)) * sqrt(223) / (4*sqrt(3) * 7^(37/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282928 Expansion of Product_{k>=1} (1 - x^(7*k))^40/(1 - x^k)^41 in powers of x.

Original entry on oeis.org

1, 41, 902, 14063, 173635, 1801745, 16300739, 131814181, 969824701, 6579564585, 41587633402, 246925024493, 1386436741480, 7402293438974, 37755020009290, 184685764132377, 869379223328495, 3949788012868677, 17363552010806127
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^40/(1 - x^j)^41: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^40/(1 - x^k)^41, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)
  • PARI
    my(m=30, x='x+O('x^m)); Vec(prod(j=1,m, (1 - x^(7*j))^40/(1 - x^j)^41)) \\ G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^40/(1 - x^n)^41.
a(n) ~ exp(Pi*sqrt(494*n/21)) * sqrt(247) / (4*sqrt(3) * 7^(41/2) * n). - Vaclav Kotesovec, Nov 10 2017

A282929 Expansion of Product_{k>=1} (1 - x^(7*k))^44/(1 - x^k)^45 in powers of x.

Original entry on oeis.org

1, 45, 1080, 18285, 244260, 2733804, 26606745, 230915656, 1819708110, 13198528010, 89041203249, 563420646090, 3366705675744, 19105222953420, 103448715353372, 536621238174195, 2675953974595655, 12866398610335149, 59805282183021050, 269356649381129943, 1177903345233332970, 5010462608512204473, 20765528801742226455
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

Links

Crossrefs

Cf. A282919.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^44/(1 - x^j)^45: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
  • Maple
    N:= 30:
gN:= mul((1-x^(7*n))^44/(1-x^n)^45,n=1..N):
S:=series(gN,x,N+1):
seq(coeff(S,x,n),n=1..N); # Robert Israel, Nov 18 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^44/(1 - x^k)^45, {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))^44/(1 - x^j)^45)) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^44/(1 - x^j)^45 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^44/(1 - x^n)^45.
a(n) ~ exp(Pi*sqrt(542*n/21)) * sqrt(271) / (4*sqrt(3) * 7^(45/2) * n). - Vaclav Kotesovec, Nov 10 2017
Showing 1-10 of 13 results. Next

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