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 1-3 of 3 results.

A327682 Expansion of Product_{k>0} (-1+sqrt(1+4*x^k))/(2*x^k).

Original entry on oeis.org

1, -1, 1, -5, 14, -40, 122, -404, 1362, -4608, 15881, -55709, 197402, -705114, 2539282, -9210196, 33605471, -123262137, 454268676, -1681305246, 6246544735, -23288217459, 87096982499, -326680267261, 1228547420236, -4631474743422, 17499462106763, -66257720483935, 251356773101419
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Links

Crossrefs

Cf. A000108, A081362, A168491, A309867, A322204, A327683.

Programs

  • Mathematica
    m = 28; CoefficientList[Series[Product[(-1 + Sqrt[1 + 4*x^k])/(2*x^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 06 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (-1+sqrt(1+4*x^k))/(2*x^k)))
  • PARI
    N=66; x='x+O('x^N); Vec(prod(i=1, N, sum(j=0, N\i, (-1)^j*binomial(2*j, j)*x^(i*j)/(j+1))))

Formula

a(n) ~ (-1)^n * c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (-1 + sqrt(1 + 4*(-1/4)^k)) / (2*(-1/4)^k) = 0.5396673413761086071059510679780476790558662471136055... - Vaclav Kotesovec, May 06 2021

A309867 Expansion of Product_{k>0} (1+sqrt(1-4*x^k))/2.

Original entry on oeis.org

1, -1, -2, -2, -5, -9, -36, -104, -365, -1219, -4213, -14617, -51570, -183084, -656536, -2370066, -8613590, -31478538, -115632718, -426676244, -1580878746, -5878933054, -21936060630, -82100980070, -308146839623, -1159545407027, -4373730398473, -16533813947503
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Links

Crossrefs

Convolution inverse of A322204.
Cf. A115140, A327682, A327683.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1-4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1-4*x^k))/2))
  • PARI
    N=66; x='x+O('x^N); Vec(prod(i=1, N, 1-sum(j=1, N\i, binomial(2*j-2, j-1)*x^(i*j)/j)))

Formula

a(n) ~ -c * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + sqrt(1 - 4*(1/4)^k))/2 = 0.4567034206737725013365271429022657551331606541289778092649... - Vaclav Kotesovec, May 06 2021

A327684 Expansion of Product_{k>0} (1 + x^k/(1 + x^k/(1 + x^k))).

Original entry on oeis.org

1, 1, 0, 4, -4, 11, -13, 39, -73, 144, -256, 559, -1116, 2188, -4317, 8804, -17591, 34992, -69815, 140097, -280416, 560077, -1119327, 2240719, -4482527, 8961129, -17920037, 35847885, -71699202, 143384383, -286760131, 573549105, -1147115913, 2294173485, -4588309651, 9176739373
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Links

Crossrefs

Cf. A000009, A268498, A327683.

Programs

  • Mathematica
    m = 35; CoefficientList[Series[Product[(1 + x^k/(1 + x^k/(1 + x^k))), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 06 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+3*x^k+x^(2*k))/(1+2*x^k)))

Formula

a(n) ~ -(-1)^n * c * 2^n, where c = 1/4 * Product_{k>=2} (1 + (-1/2)^k/(1 + (-1/2)^k/(1 + (-1/2)^k))) = 0.267077782295890034289082591596560646781284184591415208072736792505213482... - Vaclav Kotesovec, May 06 2021
Showing 1-3 of 3 results.

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