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.

Previous Showing 11-14 of 14 results.

A263414 Expansion of Product_{k>=1} 1/(1-x^(3*k+4))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 4, 2, 0, 5, 6, 1, 6, 10, 2, 7, 19, 6, 9, 28, 14, 11, 44, 28, 16, 61, 52, 25, 87, 93, 45, 116, 153, 77, 160, 244, 141, 215, 376, 244, 301, 560, 422, 422, 817, 695, 617, 1173, 1132, 917, 1661, 1776, 1399, 2331
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

Comments

In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then
a(n) ~ d3(v) * 3^(v^2/27 - 8/9) * exp(-Pi^4 * v^2 / (3888*Zeta(3)) - v * Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) * n^(v^2/54 - 25/36) / (sqrt(Pi) * 2^(v^2/54 + 11/36) * Zeta(3)^(v^2/54 - 7/36)), where
d3(v) = exp(Integral_{x=0..infinity} (exp((3-v)*x) / (exp(3*x)-1)^2 + (1/12 - v^2/18)/exp(x) - 1/(9*x^2) + v/(9*x))/x dx).
if mod(v,3)=1, then d3(v) = exp(A263031) * 2^((v+2)/6) * 3^((v+2)/18) * Pi^((v+2)/6) / (Gamma(1/3)^((v+2)/3) * A263416((v-1)/3)).
if mod(v,3)=2, then d3(v) = exp(A263030) * 2^((v+1)/6) * Pi^((v+1)/6) / (3^((v+1)/18) * Gamma(2/3)^((v+1)/3) * A263417((v-2)/3)).

Links

Crossrefs

Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263415 (v=5), A263031, A263416.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-3, 3, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 17 2015
  • Mathematica
    nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+4))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 80; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(3*k))^2)).
a(n) ~ c * exp(-Pi^4/(243*Zeta(3)) - 4*Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (sqrt(Pi) * 2^(65/108) * 3^(8/27) * Zeta(3)^(11/108) * n^(43/108)), where c = exp(A263031) * 2 * 3^(1/3) * Pi / Gamma(1/3)^2 = 1.24446091929106216111829684663735422946506...

A263415 Expansion of Product_{k>=1} 1/(1-x^(3*k+5))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 1, 4, 0, 2, 5, 0, 6, 6, 1, 10, 7, 2, 19, 8, 6, 28, 10, 14, 44, 12, 28, 60, 17, 52, 86, 26, 93, 112, 46, 152, 152, 78, 243, 196, 142, 372, 264, 244, 552, 350, 422, 798, 486, 692, 1136, 680, 1125, 1582, 997, 1758
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

Comments

In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then
a(n) ~ d3(v) * 3^(v^2/27 - 8/9) * exp(-Pi^4 * v^2 / (3888*Zeta(3)) - v * Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) * n^(v^2/54 - 25/36) / (sqrt(Pi) * 2^(v^2/54 + 11/36) * Zeta(3)^(v^2/54 - 7/36)), where
d3(v) = exp(Integral_{x=0..infinity} (exp((3-v)*x) / (exp(3*x)-1)^2 + (1/12 - v^2/18)/exp(x) - 1/(9*x^2) + v/(9*x))/x dx).
if mod(v,3)=1, then d3(v) = exp(A263031) * 2^((v+2)/6) * 3^((v+2)/18) * Pi^((v+2)/6) / (Gamma(1/3)^((v+2)/3) * A263416((v-1)/3)).
if mod(v,3)=2, then d3(v) = exp(A263030) * 2^((v+1)/6) * Pi^((v+1)/6) / (3^((v+1)/18) * Gamma(2/3)^((v+1)/3) * A263417((v-2)/3)).

Links

Crossrefs

Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263414 (v=4), A263030, A263417.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-3, 3, 'r')=2, d*r, 0)
        , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 17 2015
  • Mathematica
    nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+5))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 80; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^(3*k))^2)).
a(n) ~ c * 3^(1/27) * exp(-25*Pi^4 / (3888*Zeta(3)) - 5*Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (sqrt(Pi) * 2^(83/108) * Zeta(3)^(29/108) * n^(25/108)), where c = exp(A263030) * Pi / (3^(1/3) * Gamma(2/3)^2) = 0.98365214791227284535715328899346961376609...

A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 22, 40, 65, 107, 176, 282, 448, 705, 1101, 1701, 2611, 3977, 6021, 9048, 13527, 20102, 29720, 43712, 63997, 93259, 135317, 195539, 281440, 403559, 576568, 820888, 1164826, 1647583, 2323169, 3266041, 4578305, 6399990, 8922389, 12406535
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 15 2015

Keywords

Links

Crossrefs

Cf. A003105, A026007, A262876, A262877, A262878, A262879, A263346.
Cf. A285289, A285290, A285291.

Programs

  • Mathematica
    nmax=40; CoefficientList[Series[Product[((1 + x^k)/(1 + x^(3*k)))^k,{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ Zeta(3)^(1/6) * exp(2^(-1/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/6) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A263346 Expansion of Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 5, 12, 21, 40, 71, 130, 221, 387, 648, 1095, 1800, 2964, 4792, 7730, 12301, 19510, 30619, 47859, 74179, 114469, 175427, 267684, 406039, 613325, 921671, 1379500, 2055313, 3050652, 4509385, 6641966, 9746452, 14254242, 20775255, 30184451, 43715711
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 15 2015

Keywords

Links

Crossrefs

Cf. A000726, A000219, A262876, A262877, A262878, A262879, A263345.

Programs

  • Mathematica
    nmax=40; CoefficientList[Series[Product[((1 - x^(3*k))/(1 - x^k))^k,{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ 2^(1/6) * Zeta(3)^(1/6) * exp(6^(1/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(11/12) * sqrt(Pi) * n^(2/3)).
Previous Showing 11-14 of 14 results.

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