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.

A294750 Expansion of Product_{k>=1} 1/(1 - x^(2*k - 1))^(k^2).

Original entry on oeis.org

1, 1, 1, 5, 5, 14, 24, 40, 76, 121, 230, 356, 635, 1024, 1709, 2820, 4510, 7430, 11712, 19007, 29800, 47490, 74261, 116385, 181423, 280696, 434956, 666970, 1025816, 1562504, 2383916, 3611493, 5467505, 8241296, 12389888, 18581326, 27765501, 41426994, 61573390
Offset: 0

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Author

Vaclav Kotesovec, Nov 08 2017

Keywords

Comments

In general, if g.f. = Product_{k>=1} 1/(1 - x^(2*k-1))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(2*Pi/3 * (2*c2/15)^(1/4) * n^(3/4) + (c1+c2) * Zeta(3) / Pi^2 * sqrt(15*n/(2*c2)) + (Pi*(4*c0 + 2*c1 + c2)/24 - 15*(c1+c2)^2 * Zeta(3)^2 / (2*c2*Pi^5)) * (15*n/(2*c2))^(1/4) + 75*(c1+c2)^3 * Zeta(3)^3 / (c2^2 * Pi^8) - (5*c0 + 15*c1/4 + c2/2 + 5*c1*(2*c0 + c1) / (2*c2)) * (Zeta(3) / (4*Pi^2)) - (c1+c2)/24) * A^((c1+c2)/2) * (15/c2)^((c1+c2)/96 - 1/8) * n^((c1+c2)/96 - 5/8) / (2^(15/8 + c0/2 + (29*c1 + 17*c2)/96) * Pi^((c1+c2)/24)), where A is the Glaisher-Kinkelin constant A074962.

Links

Crossrefs

Cf. A023871, A035528, A294749, A294755.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2*Pi/3 * (2/15)^(1/4) * n^(3/4) + Zeta(3) * sqrt(15*n/2) / Pi^2 + (Pi * (15/2)^(1/4)/24 - Zeta(3)^2 * (15/2)^(5/4) / Pi^5) * n^(1/4) + 75*Zeta(3)^3 / Pi^8 - Zeta(3) / (8*Pi^2) - 1/24) * sqrt(A) / (2^(197/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, -1, 5, -14, 36, -97, 246, -593, 1423, -3351, 7699, -17432, 38901, -85545, 185862, -399220, 848080, -1783682, 3716584, -7675916, 15722127, -31951330, 64452707, -129102947, 256876062, -507854808, 997954125, -1949631802, 3787674152, -7319306458, 14071371173
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^2, g(n) = -1.

Links

Crossrefs

Product_{k>=1} (1+x^k)^((-1)^k*k^b): A083365 (b=0), A284474 (b=1), this sequence (b=2).
Cf. A027998, A177155, A266964, A294749, A294750, A307460.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 + x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^(4*k^2) / (1 + x^(2*k - 1))^((2*k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+x^k)^((-1)^k*k^2)))

Formula

a(n) ~ (-1)^n * exp(2*Pi*n^(3/4)/3 + 3*Zeta(3)/(4*Pi^2)) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 09 2019

A294755 Expansion of Product_{k>=1} ((1 + x^(2*k - 1))/(1 - x^(2*k - 1)))^(k^2).

Original entry on oeis.org

1, 2, 2, 10, 18, 36, 86, 150, 326, 608, 1164, 2230, 4046, 7632, 13622, 24868, 44222, 78304, 138312, 240138, 418648, 718292, 1233494, 2097350, 3552370, 5987642, 10026088, 16745600, 27779030, 45970868, 75650248, 124100970, 202720814, 329909400, 535132036
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2017

Keywords

Comments

Convolution of A294749 and A294750.
In general, if g.f. = Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k-1)))^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(Pi*sqrt(2) * c2^(1/4) * n^(3/4) / 3 + 7*(c1+c2) * Zeta(3) * sqrt(n) / (2*sqrt(c2) * Pi^2) + (Pi*(4*c0 + 2*c1 + c2) / (8*sqrt(2) * c2^(1/4)) - 49*(c1+c2)^2 * Zeta(3)^2 / (2^(3/2) * c2^(5/4) * Pi^5)) * n^(1/4) - (7*c0 + 21*c1/4 + c2 + 7*c0*c1/c2 + 7*c1^2/(2*c2)) * Zeta(3) / (4*Pi^2) + 22411*(c1+c2)^3 * Zeta(3)^3 / (196 * c2^2 * Pi^8) - (c1+c2)/24) * A^((c1+c2)/2) * (n^((c1+c2)/96 - 5/8) / (2^(c0/2 + (11*c1 + 5*c2)/48 + 9/4) * Pi^((c1+c2)/24) * c2^((c1+c2)/96 - 1/8))), where A is the Glaisher-Kinkelin constant A074962.

Links

Crossrefs

Cf. A206622, A292037, A294749, A294750.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(sqrt(2)*Pi * n^(3/4)/3 + 7*Zeta(3) * sqrt(n) / (2*Pi^2) + (Pi / (8*sqrt(2)) - 49*Zeta(3)^2 / (2^(3/2) * Pi^5)) * n^(1/4) + 22411*Zeta(3)^3 / (196*Pi^8) - Zeta(3)/(4*Pi^2) - 1/24) * sqrt(A) / (2^(113/48) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A294777 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 0, 6, 3, 10, 9, 15, 28, 24, 60, 47, 126, 99, 227, 225, 414, 498, 717, 1044, 1301, 2082, 2364, 3984, 4482, 7353, 8513, 13287, 16317, 23698, 30789, 42081, 57499, 74763, 105276, 133273, 190155, 238122, 338291, 425775, 596142, 759651, 1041498
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2017

Keywords

Crossrefs

Cf. A027998, A027999, A263140, A294749.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^(2*k-1))^(k*(k-1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*14^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi*5^(1/4) * n^(1/4) / (2^(17/4) * 3^(3/4) * 7^(1/4))) * 7^(1/8) / (2^(19/8) * 15^(1/8) * n^(5/8)).
Showing 1-4 of 4 results.

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