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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A022633 Expansion of Product_{m>=1} (1 + m*q^m)^5.

Original entry on oeis.org

1, 5, 20, 75, 240, 726, 2075, 5620, 14645, 36875, 90057, 214065, 497170, 1129670, 2517425, 5512125, 11871310, 25184930, 52686885, 108786970, 221894842, 447455885, 892609420, 1762608545, 3447282925, 6680871925, 12835968690, 24459374345, 46243132855, 86773966825, 161664667295
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=5 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^5:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 16 2018
  • Mathematica
    With[{nmax=50}, CoefficientList[Series[Product[(1+k*q^k)^5, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 16 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^5)) \\ G. C. Greubel, Feb 16 2018

Formula

G.f.: exp(5*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022634 Expansion of Product_{m>=1} (1 + m*q^m)^6.

Original entry on oeis.org

1, 6, 27, 110, 387, 1266, 3896, 11340, 31629, 84992, 221028, 558450, 1375615, 3310764, 7803069, 18044374, 40998078, 91653990, 201842383, 438312534, 939439674, 1988944070, 4162521165, 8617025112, 17655688602, 35823617658, 72015578091, 143499705550, 283544586489, 555779906772
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=6 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^6:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^6, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^6)) \\ G. C. Greubel, Feb 17 2018

Formula

G.f.: exp(6*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022635 Expansion of Product_{m>=1} (1 + m*q^m)^7.

Original entry on oeis.org

1, 7, 35, 154, 588, 2065, 6790, 21071, 62447, 177863, 489279, 1305402, 3389603, 8587999, 21280436, 51674728, 123161500, 288539664, 665292642, 1511359766, 3386065697, 7488093282, 16357998447, 35324428405, 75453678433, 159512035137, 333918915120, 692516812176, 1423479123640
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=7 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^7:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^7, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^7)) \\ G. C. Greubel, Feb 17 2018

Formula

G.f.: exp(7*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022636 Expansion of Product_{m>=1} (1 + m*q^m)^8.

Original entry on oeis.org

1, 8, 44, 208, 854, 3200, 11176, 36752, 115089, 345600, 1000484, 2804544, 7639718, 20280672, 52593032, 133509840, 332340788, 812455304, 1953140484, 4622589504, 10782030284, 24807035200, 56345836888, 126438750160, 280490520517, 615512622608, 1336825948592, 2875079590304
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=8 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^8:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^8, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^8)) \\ G. C. Greubel, Feb 17 2018

Formula

G.f.: exp(8*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022637 Expansion of Product_{m>=1} (1 + m*q^m)^9.

Original entry on oeis.org

1, 9, 54, 273, 1197, 4761, 17577, 60957, 200799, 633007, 1920510, 5633667, 16037700, 44439840, 120165858, 317762553, 823240341, 2092864401, 5228118701, 12848849154, 31100190048, 74208885351, 174708121455, 406132690635, 932871440739, 2118595079790, 4759875472491
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=9 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^9:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^9, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^9)) \\ G. C. Greubel, Feb 17 2018

Formula

G.f.: exp(9*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022638 Expansion of Product_{m>=1} (1 + m*q^m)^10.

Original entry on oeis.org

1, 10, 65, 350, 1630, 6852, 26635, 97030, 334990, 1104730, 3500740, 10710950, 31763985, 91589730, 257459110, 707115814, 1901162925, 5011993330, 12974420315, 33021646490, 82723179433, 204175881220, 496953703885, 1193736868990, 2832017802500, 6639914803684
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=10 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^10:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Maple
    [seq(coeff(series(mul((1+m*q^m)^(10), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^10, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^10)) \\ G. C. Greubel, Feb 17 2018

A022639 Expansion of Product_{m>=1} (1 + m*q^m)^11.

Original entry on oeis.org

1, 11, 77, 440, 2167, 9592, 39127, 149237, 538329, 1851674, 6111171, 19448573, 59922709, 179331603, 522723740, 1487454914, 4140279660, 11292030255, 30221623905, 79475723767, 205600559461, 523762010695, 1315113742769, 3257405396388, 7964974336693, 19239590761567
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=11 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^11:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Maple
    [seq(coeff(series(mul((1+m*q^m)^(11), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^11, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^11)) \\ G. C. Greubel, Feb 17 2018

A022640 Expansion of Product_{m>=1} (1 + m*q^m)^12.

Original entry on oeis.org

1, 12, 90, 544, 2823, 13116, 55982, 222936, 838011, 2998896, 10282986, 33959016, 108458924, 336141084, 1013801700, 2982628712, 8577246237, 24152726184, 66699488360, 180885417408, 482312100000, 1265779076680, 3272696917782, 8343402502128, 20989675199987, 52143220175940
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=12 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^12:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Maple
    [seq(coeff(series(mul((1+m*q^m)^(12), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^12, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^12)) \\ G. C. Greubel, Feb 17 2018

A022641 Expansion of Product_{m>=1} (1 + m*q^m)^13.

Original entry on oeis.org

1, 13, 104, 663, 3614, 17576, 78299, 324766, 1269242, 4715204, 16762551, 57327556, 189418658, 606787572, 1890046210, 5738539729, 17019191579, 49394158541, 140507716414, 392299039821, 1076369417474, 2905414115877, 7722941644821, 20233362612424, 52288914446548, 133389316899462
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=13 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^13:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Maple
    [seq(coeff(series(mul((1+m*q^m)^(13), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^13, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^13)) \\ G. C. Greubel, Feb 17 2018

A022642 Expansion of Product_{m>=1} (1 + m*q^m)^14.

Original entry on oeis.org

1, 14, 119, 798, 4557, 23142, 107366, 462856, 1876952, 7224714, 26579063, 93966992, 320651170, 1059923690, 3404112479, 10649329250, 32521525967, 97132069090, 284187808429, 815681830796, 2299630643723, 6375380037894, 17398106046384, 46777705917502, 124014391872203, 324432027054226
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=14 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^14:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Maple
    [seq(coeff(series(mul((1+m*q^m)^(14), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^14, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^14)) \\ G. C. Greubel, Feb 17 2018
Previous Showing 11-20 of 27 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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