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.

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A022582 Expansion of Product_{m>=1} (1+x^m)^17.

Original entry on oeis.org

1, 17, 153, 986, 5134, 22967, 91528, 332741, 1121864, 3550518, 10644516, 30446116, 83554915, 221028152, 565733446, 1405559677, 3398860779, 8018057345, 18489507853, 41750241112, 92455892640, 201066321781, 429927351485, 904832464581, 1876192580514, 3836193955660, 7740691696577
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=17 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^17:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^17,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^17)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (17/3)^(1/4) * exp(Pi * sqrt(17*n/3)) / (1024 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (17/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017

Extensions

More terms added by G. C. Greubel, Feb 25 2018

A022583 Expansion of Product_{m>=1} (1+x^m)^18.

Original entry on oeis.org

1, 18, 171, 1158, 6309, 29430, 121962, 460008, 1605996, 5254334, 16260867, 47949804, 135509922, 368764290, 970099191, 2475106170, 6141671649, 14856839874, 35107961175, 81189855828, 184033842021, 409446105486, 895231350108, 1925717858910, 4079428991751, 8518121246538
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=18 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^18:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^18,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^18)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (3/2)^(1/4) * exp(Pi * sqrt(6*n)) / (1024 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (18/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017

A022584 Expansion of Product_{m>=1} (1+x^m)^19.

Original entry on oeis.org

1, 19, 190, 1349, 7676, 37278, 160417, 626924, 2263698, 7647652, 24405633, 74120672, 215505334, 602763220, 1628328880, 4262845643, 10845598563, 26882001287, 65048680364, 153950675585, 356936640088, 811869015895, 1813912504439, 3985419541978, 8619872682020, 18369414409148
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=19 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^19:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^19,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^19)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (19/3)^(1/4) * exp(Pi * sqrt(19*n/3)) / (2048 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (19/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017

A022585 Expansion of Product_{m>=1} (1+x^m)^20.

Original entry on oeis.org

1, 20, 210, 1560, 9255, 46724, 208510, 843320, 3145855, 10963160, 36042250, 112633760, 336622160, 966897820, 2680139300, 7193849624, 18752326235, 47590579080, 117840608100, 285228791880, 675978772326, 1570897356960, 3584273539170, 8038904002760, 17741382028085, 38563932406500
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=20 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^20:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^20,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^20)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (5/3)^(1/4) * exp(2 * Pi * sqrt(5*n/3)) / (2048 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (20/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017

A022586 Expansion of Product_{m>=1} (1+x^m)^21.

Original entry on oeis.org

1, 21, 231, 1792, 11067, 58002, 268093, 1120899, 4315269, 15497986, 52441347, 168487473, 517184185, 1524390777, 4332440454, 11914441196, 31798680774, 82574231187, 209091601271, 517272712845, 1252351944165, 2971700764941, 6920411525727, 15835150526244, 35640093688017
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=21 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^21:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^21,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^21)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ 7^(1/4) * exp(Pi * sqrt(7*n)) / (4096 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (21/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017

A022587 Expansion of Product_{m>=1} (1 + x^m)^22.

Original entry on oeis.org

1, 22, 253, 2046, 13134, 71368, 341275, 1473494, 5848810, 21628002, 75261384, 248403586, 782547909, 2365168542, 6887441198, 19393122562, 52959869787, 140631776582, 363943223941, 919706094494, 2273411319069, 5505315501136, 13078268135683, 30514651732686, 70005101272876
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=22 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^22:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^22,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^22)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (11/6)^(1/4) * exp(Pi * sqrt(22*n/3)) / (4096 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (22/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017

A022588 Expansion of Product_{m>=1} (1 + x^m)^23.

Original entry on oeis.org

1, 23, 276, 2323, 15479, 87101, 430445, 1917349, 7839849, 29824583, 106646308, 361327079, 1167406906, 3615602714, 10780913004, 31061653709, 86741652761, 235404301651, 622271232287, 1605432041576, 4049617772390, 10002785010369, 24227747380447, 57613905606273, 134662398395411
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=23 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^23:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^23,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^23)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (23/3)^(1/4) * exp(Pi * sqrt(23*n/3)) / (8192 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (23/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017

A022589 Expansion of Product_{m>=1} (1 + q^m)^25.

Original entry on oeis.org

1, 25, 325, 2950, 21100, 126905, 667850, 3157725, 13667175, 54900675, 206841715, 736953800, 2499500175, 8113694575, 25320834800, 76253908740, 222308896150, 629146702350, 1732518057650, 4651937973250, 12201443983695, 31311905220800, 78732034002275, 194220161393825
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=25 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^25:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^25,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^25)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ sqrt(5) * exp(5 * Pi * sqrt(n/3)) / (16384 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

Extensions

Terms a(20) onward added by G. C. Greubel, Feb 25 2018

A022590 Expansion of Product_{m>=1} (1+q^m)^26.

Original entry on oeis.org

1, 26, 351, 3302, 24427, 151658, 822484, 4001660, 17799041, 73391968, 283542740, 1034983222, 3593364255, 11931569028, 38062054017, 117095671862, 348538604492, 1006539781078, 2827014674081, 7738495452714, 20683325376064, 54066855041446, 138427417637249, 347584258977384
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=26 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^26:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 19 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^26,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^26)) \\ G. C. Greubel, Feb 19 2018

Formula

a(n) ~ (13/6)^(1/4) * exp(Pi * sqrt(26*n/3)) / (16384 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

A022591 Expansion of Product_{m>=1} (1+q^m)^27.

Original entry on oeis.org

1, 27, 378, 3681, 28134, 180144, 1005957, 5032422, 22986801, 97229361, 384953553, 1438738443, 5110502256, 17348445108, 56541857409, 177611637141, 539501563962, 1589134470966, 4550281700055, 12692702415312, 34556103662778, 91975719684573, 239686155975618
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=27 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^27:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 19 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^27,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^27)) \\ G. C. Greubel, Feb 19 2018

Formula

a(n) ~ sqrt(3) * exp(3 * Pi * sqrt(n)) / (32768 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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