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 1-6 of 6 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

A370713 a(n) = 2^n * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/2).

Original entry on oeis.org

1, 2, 10, 36, 150, 524, 2116, 7528, 29158, 107404, 409388, 1526712, 5843580, 22048056, 84476488, 322465104, 1240657798, 4769404556, 18436257820, 71280387736, 276583603220, 1074167182184, 4181581140280, 16295088132272, 63609310195164, 248542739388472, 972391744910392
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

Crossrefs

Cf. A070933, A370709.

Programs

  • Mathematica
    nmax = 35; CoefficientList[Series[Product[1/(1-2*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 35; CoefficientList[Series[Product[1/(1-2*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Sqrt[-1/QPochhammer[2,x]], {x, 0, nmax}], x] * 2^Range[0, nmax]

Formula

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

A370736 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/4).

Original entry on oeis.org

1, 2, 2, 76, -106, 1788, -1516, 57176, -276634, 2270444, -10094212, 97699752, -664173444, 4819718488, -33236872088, 259931360688, -1894783205754, 13983087008588, -103270227527444, 779496572387208, -5855545477963244, 44016069418771976, -331519650617078376, 2514477954420678352
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

Cf. A032302 (m=1), A370709 (m=2), A370716 (m=3), A370737 (m=5).

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[1+2*x^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[1+2*(4*x)^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + 2*(4*x)^k)^(1/4).
a(n) ~ (-1)^(n+1) * QPochhammer(-1/2)^(1/4) * 8^n / (4 * Gamma(3/4) * n^(5/4)).

A370739 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, -75, 35250, -1138125, 72645000, -3307996875, 244578890625, -15502648125000, 985908809765625, -63515254624218750, 4314500023927734375, -291905297026816406250, 19789483493484814453125, -1355414138248614990234375, 93666904586649390380859375, -6498800175020013123779296875
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 + d*x^k)^(1/m), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d)^(1/m) * d^n / (m*Gamma(1 - 1/m) * n^(1 + 1/m)).

Crossrefs

Cf. A032308 (d=3,m=1), A370711 (d=3,m=2), A370712 (d=3,m=3), A370738 (d=3,m=4).
Cf. A032302 (d=2,m=1), A370709 (d=2,m=2), A370716 (d=2,m=3), A370736 (d=2,m=4), A370737 (d=2,m=5).
Cf. A000009 (d=1,m=1), A298994 (d=1,m=2), A303074 (d=1,m=3)

Programs

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

Formula

G.f.: Product_{k>=1} (1 + 3*(25*x)^k)^(1/5).
a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/5) * 75^n / (5 * Gamma(4/5) * n^(6/5)).

A370737 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/5).

Original entry on oeis.org

1, 10, 50, 14750, -166250, 14011250, -133418750, 18136968750, -620089531250, 29520532031250, -917207280468750, 51260806902343750, -2257145499863281250, 101035630688769531250, -4434459153208496093750, 214279556679692871093750, -9859289197933918457031250, 454976266920750451660156250
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

Cf. A032302 (m=1), A370709 (m=2), A370716 (m=3), A370736 (m=4).

Programs

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

Formula

G.f.: Product_{k>=1} (1 + 2*(25*x)^k)^(1/5).
a(n) ~ (-1)^(n+1) * QPochhammer(-1/2)^(1/5) * 50^n / (5 * Gamma(4/5) * n^(6/5)).

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/2).

Original entry on oeis.org

1, 10, 62, 1620, 6966, 157580, 1284012, 19189160, 73908774, 2233414620, 9656822916, 287668788120, -324007115716, 40151699854200, -199460032590312, 7130611518222160, -64971542557275642, 1292318115470489340, -15433712240157937260, 265667290368470451000, -3624776372747687578668
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 01 2024

Keywords

Comments

In general, if d > 1 and g.f. = Product_{k>=1} ((1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k))^(1/2), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d) * d^(2*n) / (2*sqrt((1 + 1/d)*Pi) * n^(3/2)).

Crossrefs

Cf. A032302, A304961, A370709, A370761, A370765.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(3*k+1)*x^k)*(1+2^(3*k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x])^(1/2)/3, {x, 0, nmax}], x] * 8^Range[0, nmax]

Formula

G.f.: Product_{k>=1} ((1 + 2^(3*k+1)*x^k) * (1 + 2^(3*k-1)*x^k))^(1/2).
a(n) ~ (-1)^(n+1) * c * 16^n / n^(3/2), where c = QPochhammer(-1/2) / sqrt(6*Pi) = 0.278865402428524528968820654198674...
Showing 1-6 of 6 results.

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