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.

A285070 Expansion of Product_{k>=0} (1-x^(4*k+1))^(4*k+1).

Original entry on oeis.org

1, -1, 0, 0, 0, -5, 5, 0, 0, -9, 19, -10, 0, -13, 58, -55, 10, -17, 118, -191, 95, -26, 223, -512, 400, -116, 362, -1175, 1329, -564, 609, -2368, 3593, -2218, 1246, -4402, 8600, -7118, 3433, -7792, 18503, -19778, 10702, -13924, 37009, -49017, 32097, -27141
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+1))^(m*k+1): A285069 (m=2), A285050 (m=3), this sequence (m=4), A285071 (m=5).
Cf. A285048, A285288.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^(4*k-3))^(4*k-3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 17 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

A285050 Expansion of Product_{k>=0} (1-x^(3*k+1))^(3*k+1).

Original entry on oeis.org

1, -1, 0, 0, -4, 4, 0, -7, 13, -6, -10, 38, -32, -9, 74, -103, 27, 137, -266, 153, 191, -593, 537, 167, -1161, 1437, -222, -2035, 3397, -1578, -3110, 7160, -5285, -3712, 13942, -13920, -2002, 24848, -32241, 6764, 40661, -68059, 32487, 59109, -133506, 95221, 71243
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+1))^(m*k+1): A285069 (m=2), this sequence (m=3), A285070 (m=4), A285071 (m=5).
Cf. A262947.

A285049 Expansion of Product_{k>=0} 1/(1-x^(5*k+1))^(5*k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 18, 39, 39, 39, 39, 55, 121, 177, 177, 177, 198, 360, 591, 717, 717, 743, 1045, 1777, 2393, 2645, 2676, 3199, 4982, 7264, 8650, 9148, 9956, 13760, 20348, 26060, 28873, 30869, 38134, 54634, 73142, 85536, 92302, 106501, 143167
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-m+1))^(m*k-m+1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(19*m/36 + 1/(6*m) - 1) * m^(17*m/36 + 5/(6*m) - 3/2) * Pi^(m/2 - 1) * Zeta(3)^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1/m)^(m-1) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((2*m-1)*x) / (x*(exp(m*x) - 1)^2) + (1/12 - (m-1)^2/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - (m-1)/(m^2*x^2) dx. - Vaclav Kotesovec, Apr 17 2017

Links

Crossrefs

Product_{k>=0} 1/(1-x^(m*k+1))^(m*k+1): A000219 (m=1), A262811 (m=2), A262947 (m=3), A285048 (m=4), this sequence (m=5).
Cf. A285071.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1-x^(5*k-4))^(5*k-4), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

Formula

a(n) ~ 2^(301/180) * 5^(37/36) * Pi^(3/2) * Zeta(3)^(31/180) * exp(5*c + 3 * 2^(-2/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) / (sqrt(3) * Gamma(1/5)^4 * n^(121/180)), where c = Integral_{x=0..inf} ((-71/(exp(x)*300) + 1/(exp(x)*(1 - exp(-5*x))^2) - 1/(25*x^2) - 4/(25*x))/x) dx = 0.186382690624752630391368364629918483384424086341764409146923686... - Vaclav Kotesovec, Apr 16 2017

A285338 Expansion of Product_{k>=1} (1 + x^(5*k-4))^(5*k-4).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 6, 6, 0, 0, 0, 11, 26, 15, 0, 0, 16, 82, 86, 20, 0, 21, 172, 316, 180, 15, 26, 328, 872, 790, 226, 37, 538, 2043, 2681, 1310, 202, 845, 4184, 7426, 5390, 1447, 1290, 7855, 18067, 17705, 7277, 2662, 13723, 39468, 50030, 28707, 8742, 22979, 79760
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 17 2017

Keywords

Comments

For all n<=30 a(n) = abs(A285071(n)), but a(31) <> abs(A285071(31)).
In general, if m >= 1 and g.f. = Product_{k>=1} (1 + x^(m*k-m+1))^(m*k-m+1), then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(1/6 + 1/(2*m) + m/12) * 3^(1/3) * m^(1/6) * sqrt(Pi) * n^(2/3)).

Links

Crossrefs

Product_{k>=0} (1 + x^(m*k+1))^(m*k+1): A026007 (m=1), A262736 (m=2), A262949 (m=3), A285288 (m=4), this sequence (m=5).
Cf. A285071, A285340.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^(5*k-4))^(5*k-4), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2^(-4/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(41/60) * 3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)).
Showing 1-4 of 4 results.

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