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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A022570 Expansion of Product_{m>=1} (1+x^m)^5.

Original entry on oeis.org

1, 5, 15, 40, 95, 206, 425, 835, 1575, 2880, 5121, 8885, 15095, 25165, 41240, 66562, 105945, 166480, 258560, 397235, 604162, 910325, 1359680, 2014235, 2961000, 4321283, 6263360, 9019555, 12908945, 18367805, 25990149, 36581200, 51228175, 71393555, 99037095, 136775685, 188091960
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000009.
Column k=5 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^5:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^5,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^5)) \\ Indranil Ghosh, Apr 03 2017

Formula

a(n) ~ (5/3)^(1/4) * exp(Pi * sqrt(5*n/3)) / (16 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A022572 Expansion of Product_{m>=1} (1+x^m)^7.

Original entry on oeis.org

1, 7, 28, 91, 259, 665, 1589, 3585, 7707, 15925, 31808, 61677, 116536, 215180, 389194, 690935, 1206016, 2072700, 3511851, 5872545, 9701097, 15844866, 25606840, 40974528, 64956836, 102076289, 159084401, 245995792, 377574402, 575459136, 871189669, 1310492547, 1959326215, 2912370944
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000009.
Column k=7 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^7:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^7,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
  • PARI
    x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^7)) \\ Indranil Ghosh, Apr 03 2017

Formula

a(n) ~ (7/3)^(1/4) * exp(Pi * sqrt(7*n/3)) / (32 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (7/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(7*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A022574 Expansion of Product_{m>=1} (1+x^m)^9.

Original entry on oeis.org

1, 9, 45, 174, 576, 1701, 4614, 11709, 28125, 64525, 142353, 303552, 628251, 1266273, 2492352, 4801578, 9071973, 16837893, 30744649, 55296000, 98070633, 171683463, 296919081, 507695670, 858866880, 1438391232, 2386178649, 3923081006, 6395198049, 10341173376, 16593811467
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000009.
Column k=9 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^9:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^9,{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)^9)) \\ G. C. Greubel, Feb 26 2018

Formula

a(n) ~ 3^(1/4) * exp(Pi * sqrt(3*n)) / (64 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (9/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(9*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A022579 Expansion of Product_{m>=1} (1+x^m)^14.

Original entry on oeis.org

1, 14, 105, 574, 2576, 10052, 35273, 113794, 342699, 974176, 2635955, 6833540, 17061345, 41197422, 96544003, 220212384, 490104727, 1066552228, 2273590095, 4755188704, 9771319068, 19751596934, 39317784863, 77150246040, 149357609184, 285497384004, 539227765104, 1006978117880
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=14 of A286335. Cf. A000707, A023003.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^14: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)^14,{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)^14)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (7/6)^(1/4) * exp(Pi * sqrt(14*n/3)) / (256 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (14/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f. A(x) = (1/2)*( G(sqrt(x)) + G(-sqrt(x)) )/G(x^4), where G(x) = Product_{n >= 1} 1/(1 - x^n)^4 is the g.f. of A023003 (see also A000727). - Peter Bala, Oct 05 2023

A303071 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + x^k)^n.

Original entry on oeis.org

1, 2, 6, 23, 90, 362, 1491, 6225, 26242, 111479, 476466, 2046464, 8825559, 38191467, 165751529, 721177328, 3144703234, 13739010855, 60127642329, 263545670385, 1156732481150, 5083320593976, 22364017244278, 98491038664903, 434160710647831, 1915482295831037, 8457663096970431
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2018

Keywords

Links

Crossrefs

Cf. A036469, A270913, A286335, A303070.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - x) Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Exp[n Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

Formula

a(n) = [x^n] (1/(1 - x))*exp(n*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).
a(n) = Sum_{j=0..n} A286335(j,n).
a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = 0.44252758868364961050787771300805... - Vaclav Kotesovec, May 19 2018

A022575 Expansion of Product_{m>=1} (1+x^m)^10.

Original entry on oeis.org

1, 10, 55, 230, 815, 2562, 7360, 19700, 49755, 119700, 276278, 615130, 1326965, 2783360, 5693305, 11384326, 22299655, 42865280, 80983060, 150571340, 275840009, 498410280, 889056835, 1566896280, 2730474975, 4707724814, 8035618655, 13586253440, 22765030080, 37820087380
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=10 of A286335.
Cf. A000009.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^10:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^10,{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)^10)) \\ G. C. Greubel, Feb 26 2018

Formula

a(n) ~ (5/6)^(1/4) * exp(Pi * sqrt(10*n/3)) / (64 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (10/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

A022576 Expansion of Product_{m>=1} (1+x^m)^11.

Original entry on oeis.org

1, 11, 66, 297, 1122, 3740, 11341, 31922, 84535, 212707, 512369, 1188353, 2666048, 5807296, 12319659, 25518757, 51725289, 102786959, 200568907, 384847199, 727019260, 1353654049, 2486522369, 4509972819, 8083287432, 14326409152, 25124415635, 43622744968, 75026666913, 127882738709
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=11 of A286335.
Cf. A000009.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^11:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+q^m)^11,{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)^11)) \\ G. C. Greubel, Feb 26 2018

Formula

a(n) ~ (11/3)^(1/4) * exp(Pi * sqrt(11*n/3)) / (128 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (11/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

A022578 Expansion of Product_{m>=1} (1+x^m)^13.

Original entry on oeis.org

1, 13, 91, 468, 1989, 7384, 24739, 76427, 220948, 604175, 1575392, 3941847, 9511944, 22226049, 50458447, 111609537, 241099027, 509680951, 1056262792, 2149214288, 4299359012, 8465605408, 16424772637, 31429372312, 59365381608, 110770031489, 204315725953, 372772306309, 673125106316
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=13 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^13: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)^13,{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)^13)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (13/3)^(1/4) * exp(Pi * sqrt(13*n/3)) / (256 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (13/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

Extensions

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

A022580 Expansion of Product_{m>=1} (1+x^m)^15.

Original entry on oeis.org

1, 15, 120, 695, 3285, 13443, 49305, 165795, 519240, 1531960, 4295046, 11520000, 29718605, 74060355, 178930605, 420368858, 962785560, 2154411120, 4718952965, 10134292275, 21369644184, 44300604895, 90390209685, 181706747280, 360207189225, 704726281002, 1361748557400
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=15 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^15: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)^15,{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)^15)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ 5^(1/4) * exp(Pi * sqrt(5*n)) / (512 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (15/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

A022581 Expansion of Product_{m>=1} (1+x^m)^16.

Original entry on oeis.org

1, 16, 136, 832, 4132, 17696, 67712, 236928, 770442, 2355824, 6834240, 18940480, 50424536, 129535968, 322288128, 779022208, 1834203955, 4216133616, 9479688992, 20884408704, 45148577668, 95902505120, 200394848512, 412350614016, 836328261438, 1673337795840, 3305364030464, 6450386567104, 12443955363352, 23745951691328, 44844655553536, 83856163515776, 155331420821337
Offset: 0

Views

Author

N. J. A. Sloane, Jun 14 1998

Keywords

Links

Crossrefs

Column k=16 of A286335.

Programs

  • Magma
    Coefficients(&*[(1+x^m)^16: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)^16,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
s = (QPochhammer[-1, q]/2)^16 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
  • PARI
    q='q+O('q^66); gf=(eta(q^2)/eta(q))^16; Vec(gf) \\ Joerg Arndt, Jul 06 2011
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^16)) \\ G. C. Greubel, Feb 25 2018

Formula

Expansion of q^(-2/3)(eta(q^2)/eta(q))^16 in powers of q. - Michael Somos, Jun 06 2005
Euler transform of period 2 sequence [16, 0, ...]. - Michael Somos, Jun 06 2005
G.f.: G(x) = (Prod_{k>0} 1+x^k)^16.
Let P(x) = prod(n>=1, (1+x^n)) (the g.f. for partitions into distinct parts, A000009). Then P(x^2)^8 + 16*x*P(x^2)^16*P(x)^8 = P(x)^16 (cf. A022581). - Joerg Arndt, Jul 12 2009
a(n) ~ exp(4 * Pi * sqrt(n/3)) / (256 * sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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