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.

A263145 Expansion of Product_{k>=1} (1+x^(5*k-1))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 0, 4, 4, 0, 0, 1, 10, 5, 0, 0, 6, 16, 6, 0, 0, 14, 28, 7, 0, 3, 32, 40, 8, 0, 10, 63, 60, 9, 0, 33, 112, 80, 10, 3, 74, 187, 110, 11, 14, 161, 300, 140, 12, 46, 308, 455, 182, 14, 120, 568, 672, 224, 26, 283
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 10 2015

Keywords

Links

Crossrefs

Cf. A263146, A263147, A263148, A263141, A263140, A262878, A263138.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1+x^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2).
a(n) ~ 2^(27/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(32400*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (900*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).

A263144 Expansion of Product_{k>=1} 1/(1-x^(5*k-4))^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 6, 9, 9, 9, 9, 13, 19, 23, 23, 23, 28, 42, 51, 56, 56, 62, 84, 108, 120, 126, 133, 170, 219, 253, 268, 283, 335, 427, 503, 547, 574, 658, 815, 977, 1080, 1144, 1265, 1534, 1836, 2068, 2209, 2408, 2832, 3396, 3864, 4178, 4505
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 10 2015

Keywords

Comments

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} 1/(1 - x^(s*k-t))^k then a(n) ~ s^(t^2/(3*s^2) - 7/18) * n^(t^2/(6*s^2) - 25/36) * exp(d(s,t) - Pi^4 * t^2 / (432*s^2 * Zeta(3)) + Pi^2 * t * 2^(2/3) * s^(2/3) * n^(1/3) / (12 * s^2 * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / (2^(2/3)*s^(2/3))) / (2^(t^2/(6*s^2) + 11/36) * sqrt(3*Pi) * Zeta(3)^(t^2/(6*s^2) - 7/36)), where d(s,t) = Integral_{x=0..infinity} 1/x * (exp(-(s-t)*x)/(1 - exp(-s*x))^2 - 1/(s^2*x^2) - t/(s^2*x) + exp(-x)*(1/12 - t^2/(2*s^2))) dx.

Links

Crossrefs

Cf. A263141, A263142, A263143, A263181, A263148.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz
  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-4))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} 1/j*x^j/(1 - x^(5*j))^2).
a(n) ~ Zeta(3)^(79/900) * exp(d54 - Pi^4/(675*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (75*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(371/900) * 5^(79/450) * sqrt(3*Pi) * n^(529/900)), where d54 = A263181 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 4/(25*x^2) - 71/(300*x*exp(x)) = 0.1863826906247526303913683646299184833844240863417644... .

A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 3, 0, 4, 0, 1, 4, 0, 10, 0, 6, 5, 0, 16, 0, 14, 6, 3, 28, 0, 32, 7, 10, 40, 0, 63, 8, 33, 60, 3, 112, 9, 74, 80, 14, 187, 10, 161, 110, 46, 300, 12, 308, 140, 120, 455, 24, 568, 182, 283, 672, 54, 968, 224, 594, 963, 146
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 10 2015

Keywords

Links

Crossrefs

Cf. A263145, A263146, A263148, A263143.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1+x^(5k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(2*j)/(1 - x^(5*j))^2).
a(n) ~ 2^(43/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(3600*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (300*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).

A263146 Expansion of Product_{k>=1} (1+x^(5*k-2))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 4, 0, 4, 1, 0, 10, 0, 5, 6, 0, 16, 0, 6, 14, 0, 28, 3, 7, 32, 0, 40, 10, 8, 63, 0, 60, 33, 9, 112, 3, 80, 74, 10, 187, 14, 110, 161, 11, 300, 46, 140, 308, 13, 455, 120, 182, 568, 25, 672, 283, 224, 968, 55, 963
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 10 2015

Keywords

Links

Crossrefs

Cf. A263145, A263147, A263148, A263142, A262879.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1+x^(5k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(3*j)/(1 - x^(5*j))^2).
a(n) ~ 2^(33/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(8100*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (450*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).
Showing 1-4 of 4 results.

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