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 1-5 of 5 results.

A377833 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(-x) ).

Original entry on oeis.org

1, 4, 51, 1174, 39833, 1799136, 101821723, 6938396368, 553482404721, 50619262481920, 5223014483031491, 600332651141435136, 76075005337204547209, 10538051760153093320704, 1584264031801742560408875, 256912816791069951740348416, 44703731640012047610981808097
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(4*n-k+2, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x))^3.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(4*n-k+2,n-k)/k!.

A377810 E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^2.

Original entry on oeis.org

1, 3, 17, 154, 1993, 34066, 728209, 18733926, 564117425, 19473863986, 758421401401, 32901791851006, 1573602042306265, 82267318018246986, 4667656830688700801, 285662368622361581206, 18758565855176593500385, 1315663025587514658845026, 98160436697525045768511721
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^2))/(1-x)^2))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+k+1, n-k)/k!);

Formula

E.g.f.: exp( -LambertW(-x/(1-x)^2) )/(1-x)^2.
E.g.f.: -LambertW(-x/(1-x)^2)/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+k+1,n-k)/k!.
a(n) ~ 2^(n + 3/2) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(3/2) * (1 + 2*exp(-1) - sqrt(1 + 4*exp(-1)))^(n + 1/2) * exp(2*n+1)). - Vaclav Kotesovec, Nov 11 2024

A377828 E.g.f. satisfies A(x) = (1 + x)^3 * exp(x * A(x)).

Original entry on oeis.org

1, 4, 21, 193, 2669, 48711, 1113325, 30615019, 984983193, 36319515355, 1510538562641, 69968975169567, 3572684914283941, 199389519518767111, 12075888110164192917, 788850329621989132771, 55289606764547108653361, 4138807268239824817387443, 329564746571982961088975257
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(3*k+3, n-k)/k!);

Formula

E.g.f.: (1+x)^3 * exp( -LambertW(-x*(1+x)^3) ).
E.g.f.: -LambertW(-x*(1+x)^3)/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(3*k+3,n-k)/k!.

A377599 E.g.f. satisfies A(x) = exp( x * A(x) / (1-x)^2 ) / (1-x).

Original entry on oeis.org

1, 2, 13, 145, 2277, 46461, 1172713, 35374697, 1243296169, 49940748073, 2258238723021, 113567169318285, 6289161888870061, 380364426242671469, 24948313525570134001, 1764095427822803465521, 133782341347522663175889, 10832097536377585282160337, 932693691617428946786304661
Offset: 0

Views

Author

Seiichi Manyama, Nov 14 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))/(1-x)))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+2*k, n-k)/k!);

Formula

E.g.f.: exp( -LambertW(-x/(1-x)^3) )/(1-x).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2*k,n-k)/k!.

A377608 E.g.f. satisfies A(x) = exp( x * A(x) / (1-x) ) / (1-x)^2.

Original entry on oeis.org

1, 3, 19, 202, 3085, 61886, 1544029, 46182900, 1612759369, 64455582394, 2902794546961, 145497909334856, 8035136800888333, 484821204654219798, 31735810390729211173, 2240132583683741633116, 169624462686462529305745, 13715713402047448280358002, 1179576532854283015832748697
Offset: 0

Views

Author

Seiichi Manyama, Nov 14 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))/(1-x)^2))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+2*k+1, n-k)/k!);

Formula

E.g.f.: exp( -LambertW(-x/(1-x)^3) )/(1-x)^2.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2*k+1,n-k)/k!.
Showing 1-5 of 5 results.