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 1-5 of 5 results.

A303346 Expansion of Product_{n>=1} ((1 + 2*x^n)/(1 - 2*x^n))^(1/2).

Original entry on oeis.org

1, 2, 4, 10, 18, 38, 72, 142, 260, 510, 940, 1814, 3362, 6490, 12112, 23466, 44114, 85766, 162516, 317190, 604806, 1184682, 2271248, 4461514, 8591784, 16916490, 32696708, 64496130, 125037142, 247007142, 480077432, 949510526, 1849375796, 3661330398, 7144215452
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2018

Keywords

Links

Crossrefs

Cf. A032302, A070933, A261584, A303307, A303345, A370709, A370713.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[Sqrt[-QPochhammer[-2, x] / (3*QPochhammer[2, x])], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+2*x^k)/(1-2*x^k))^(1/2)))

Formula

a(n) ~ 2^n / sqrt(c*Pi*n), where c = A048651 * A083864 = 1/2 * Product_{j>=1} (2^j-1)/(2^j+1) = 0.06056210400129025123042464659093375290492912341... - Vaclav Kotesovec, Apr 22 2018

A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

Original entry on oeis.org

1, 2, 2, 20, 6, 108, 148, 776, -186, 5964, -4, 51032, -89700, 512120, -1259416, 6406032, -19733434, 78363148, -268823572, 1047941688, -3800035916, 14327505832, -52766730600, 199492430192, -746479735524, 2811936761016, -10588174502568, 40092283176560, -151796846803592
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

Crossrefs

Cf. A032302, A370713.
Cf. A075900, A298994, A300581, A304961, A327550.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 2*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-2, x]/3], {x, 0, nmax}], x] * 2^Range[0, nmax]

Formula

G.f.: Product_{k>=1} (1 + 2*(2*x)^k)^(1/2).
a(n) ~ (-1)^(n+1) * c * 4^n / n^(3/2), where c = QPochhammer(-1/2)^(1/2) / (2*sqrt(Pi)) = 0.31039710860287467176143051675437...

A370732 a(n) = 4^n * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/4).

Original entry on oeis.org

1, 2, 18, 108, 822, 4796, 37492, 231704, 1738150, 11857004, 87262684, 617409128, 4638712124, 33724007896, 253800160808, 1894353653552, 14350905612038, 108412437326412, 827441075006796, 6308125533133896, 48388714839180756, 371391625244862600, 2860885559165073624
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

Cf. A070933 (m=1), A370713 (m=2), A370715 (m=3), A370733 (m=5).

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1-2*x^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[1/(1-2*(4*x)^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - 2*(4*x)^k)^(1/4).
a(n) ~ 8^n / (Gamma(1/4) * QPochhammer(1/2)^(1/4) * n^(3/4)).

A370735 a(n) = 5^(2*n) * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, 1050, 52125, 3277500, 179801250, 11966690625, 738318187500, 49788716718750, 3314446448437500, 227432073022265625, 15631633385109375000, 1090877899335878906250, 76338563689129101562500, 5384934139819611328125000, 381204340327212964599609375, 27111589537137988341064453125
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Comments

In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} 1/(1 - d*x^k)^(1/m), then a(n) ~ d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

Crossrefs

Cf. A242587 (d=3,m=1), A370714 (d=3,m=2), A370710 (d=3,m=3), A370734 (d=3,m=4).
Cf. A070933 (d=2,m=1), A370713 (d=2,m=2), A370715 (d=2,m=3), A370732 (d=2,m=4), A370733 (d=2,m=5).
Cf. A000041 (d=1,m=1), A271235 (d=1,m=2), A271236 (d=1,m=3).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-3*(25*x)^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - 3*(25*x)^k)^(1/5).
a(n) ~ 75^n / (Gamma(1/5) * QPochhammer(1/3)^(1/5) * n^(4/5)).

A370733 a(n) = 5^(2*n) * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/5).

Original entry on oeis.org

1, 10, 550, 19750, 921250, 32011250, 1563143750, 58080093750, 2719958906250, 113913469531250, 5214823539843750, 228024893230468750, 10704801509316406250, 482674223446582031250, 22664252188144042968750, 1053427002068999511718750, 49776941230938518066406250
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

Cf. A070933 (m=1), A370713 (m=2), A370715 (m=3), A370732 (m=4).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1-2*x^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-2*(25*x)^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - 2*(25*x)^k)^(1/5).
a(n) ~ 50^n / (Gamma(1/5) * QPochhammer(1/2)^(1/5) * n^(4/5)).
Showing 1-5 of 5 results.

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