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 1-4 of 4 results.

A271236 G.f.: Product_{k>=1} 1/(1 - (9*x)^k)^(1/3).

Original entry on oeis.org

1, 3, 45, 450, 5805, 52326, 705591, 6190425, 77219325, 751178610, 8522919063, 80502824835, 975122402985, 8949951461925, 100088881882830, 1003346683458480, 10828622925516312, 104307212166072165, 1152197107898173875, 11048535008792967825, 119509353627934830327
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 02 2016

Keywords

Comments

In general, for h>=1, if g.f. = Product_{k>=1} 1/(1-(h^2*x)^k)^(1/h), then a(n) ~ h^(2*n) * exp(Pi*sqrt(2*n/(3*h))) / (2^(5*h+3) * 3^(h+1) * h^(h+1) * n^(3*h+1))^(1/(4*h)).
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = 9^n. - Seiichi Manyama, Apr 20 2018

Links

Crossrefs

Cf. A298994, A303074, A303130, A303152, A303342, A370735.
Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), this sequence (b=3), A303135 (b=4), A303136 (b=5).

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1 - (9*x)^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, 1/(1-(9*x)^k)^(1/3))) \\ Altug Alkan, Apr 20 2018

Formula

a(n) ~ 3^(2*n - 2/3) * exp(sqrt(2*n)*Pi/3) / (2^(3/2) * n^(5/6)).

A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

Original entry on oeis.org

1, -2, -10, -20, -90, 132, -836, 6040, 2310, 60180, 180308, 1662568, -2995620, 24401320, 44072120, -102437328, 19390406, 2649221300, -10584460060, 14475802440, -228570333836, -815899620616, 2088529753800, -5590702681520, -100828534100580, -172013432412024
Offset: 0

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Author

William J. Keith, Jan 18 2018

Keywords

Comments

The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution.
Sequence appears to be positive and negative roughly half the time.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Links

Crossrefs

Cf. A000108, A271235, A298994.
Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), this sequence (b=2), A303152 (b=3), A303153 (b=4), A303154 (b=5).

Programs

  • Mathematica
    Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}]
  • PARI
    q='q+O('q^99); Vec(eta(4*q)^(1/2)) \\ Altug Alkan, Apr 20 2018

Formula

G.f.: Product_{k>=1} (1 - (4x)^k)^(1/2).

A303153 Expansion of Product_{n>=1} (1 - (16*x)^n)^(1/4).

Original entry on oeis.org

1, -4, -88, -992, -19360, -97152, -4296448, 4539392, -568015360, -127621120, -39357927424, 2424998313984, -38804685471744, 799759166930944, 4879962868940800, 41563181340426240, 585185165832486912, 55834295603426754560, -75535223925056208896
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/4, g(n) = 16^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), A303152 (b=3), this sequence (b=4), A303154 (b=5).
Cf. A303124, A303135.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(16*x)^k)^(1/4)))

A303154 Expansion of Product_{n>=1} (1 - (25*x)^n)^(1/5).

Original entry on oeis.org

1, -5, -175, -3250, -100625, -1015000, -58034375, -154171875, -22257500000, -154144921875, -6824828906250, 175448177734375, -8774446542968750, 164769756689453125, 756859169189453125, 9661555852294921875, -16148589271240234375, 81663068586871337890625
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/5, g(n) = 25^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), A303152 (b=3), A303153 (b=4), this sequence (b=5).
Cf. A303125, A303136.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(25*x)^k)^(1/5)))
Showing 1-4 of 4 results.

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