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.

A285213 Expansion of Product_{k>=0} (1-x^(4*k+3))^(4*k+3).

Original entry on oeis.org

1, 0, 0, -3, 0, 0, 3, -7, 0, -1, 21, -11, 0, -21, 54, -15, 7, -96, 122, -19, 74, -311, 217, -44, 351, -768, 367, -209, 1227, -1663, 591, -989, 3402, -3225, 1156, -3609, 8289, -5815, 3053, -11096, 18015, -10176, 9466, -29593, 36249, -18454, 28960, -71093, 68438
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+m-1))^(m*k+m-1): A285069 (m=2), A285212 (m=3), this sequence (m=4), A285214 (m=5).
Cf. A285131.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^(4*k-1))^(4*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
  • PARI
    x='x+O('x^100); Vec(prod(k=0, 100, (1 - x^(4*k + 3))^(4*k + 3))) \\ Indranil Ghosh, Apr 15 2017

Formula

a(n) ~ (-1)^n * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 17 2017

A285132 Expansion of Product_{k>=0} 1/(1-x^(5*k+4))^(5*k+4).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 0, 10, 9, 0, 0, 20, 36, 14, 0, 35, 90, 101, 19, 56, 180, 320, 202, 108, 315, 730, 859, 492, 533, 1390, 2300, 2139, 1354, 2393, 4835, 6475, 5098, 4619, 8813, 14926, 16395, 12982, 15751, 28962, 41162, 40256, 35200, 51731, 85365, 106145
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Comments

In general, if m > 1 and g.f. = Product_{k>=1} 1/(1-x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1 - 1/m) * m^(1/2 - 5/(6*m) + m/36) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((m+1)*x) / (x*(exp(m*x)-1)^2) + (1/12 - 1/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - 1/(m^2*x^2) dx. - Vaclav Kotesovec, Apr 17 2017

Links

Crossrefs

Product_{k>=0} 1/(1-x^(m*k+m-1))^(m*k+m-1): A262811 (m=2), A262946 (m=3), A285131 (m=4), this sequence (m=5).
Cf. A285214.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/(1-x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
  • PARI
    x='x+O('x^100); Vec(prod(k=0, 100, 1/(1 - x^(5*k + 4))^(5*k + 4))) \\ Indranil Ghosh, Apr 15 2017

Formula

a(n) ~ exp(5*c + 3*2^(-2/3)*5^(-1/3)*Zeta(3)^(1/3)*n^(2/3)) * (2*Zeta(3))^(31/180) / (sqrt(3) * 5^(17/36) * Gamma(4/5) * n^(121/180)), where c = Integral_{x=0..inf} ((19/(exp(x)*300) + 1/(exp(4*x)*(1-exp(-5*x))^2) - 1/(25*x^2) - 1/(25*x))/x) dx = -0.12699586713882325294527057473113580561183418857868946729897216431919... - Vaclav Kotesovec, Apr 15 2017

A285311 Expansion of Product_{k>=0} 1/(1 + x^(4*k+3))^(4*k+3).

Original entry on oeis.org

1, 0, 0, -3, 0, 0, 6, -7, 0, -10, 21, -11, 15, -42, 61, -36, 70, -150, 150, -124, 278, -441, 375, -468, 909, -1131, 1018, -1581, 2602, -2810, 2947, -4819, 6768, -6980, 8509, -13389, 16788, -17609, 23722, -34720, 40337, -44863, 63128, -85430, 95887, -114037
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2017

Keywords

Links

Crossrefs

Product_{k>=0} 1/(1 + x^(m*k+m-1))^(m*k+m-1): A284628 (m=2), A285310 (m=3), this sequence (m=4).
Cf. A285131.

Formula

a(n) = (-1)^n * A285131(n).
Showing 1-3 of 3 results.

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