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.

A380723 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2) / (1 - x*A(x)^2).

Original entry on oeis.org

1, 2, 21, 436, 13785, 589206, 31825381, 2080523880, 159761186577, 14097898530730, 1405926737063541, 156379679761925148, 19195200442017128425, 2577494115099820986174, 375845854490491567916805, 59145488004443221188738256, 9990898494797767848442559649, 1803160967691789114062089511250
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(3*n-k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(3*n-k,n-k)/k!.

A360609 E.g.f. satisfies A(x) = exp(x*A(x)^3) / (1-x).

Original entry on oeis.org

1, 2, 17, 313, 9053, 357941, 17975605, 1095604133, 78570635225, 6482415935449, 604889610870881, 62989604872166897, 7241672622495518773, 911048848278644776949, 124497704904842673086285, 18364053909500922198147421, 2908158473059042016441887025
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(-3*x/(1-x)^3)/(-3*x))^(1/3)))

Formula

E.g.f.: (LambertW( -3*x/(1-x)^3 ) / (-3*x))^(1/3).
a(n) ~ 3^(-5/6) * (2^(4/3) + 2*(3 + sqrt(4*exp(1) + 9))^(1/3) * exp(-2/3) - 2^(2/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3))^(1/6) * 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(4/9) * sqrt(4 - 2^(4/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + 3*2^(2/3) * exp(-2/3) * (3 + sqrt(4*exp(1) + 9))^(1/3)) * n^(n-1) * (12 + 4*sqrt(4*exp(1) + 9))^(n/3) / (exp(7/18 + 5*n/3) * (2 - 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + exp(-2/3) * (12 + 4*sqrt(4*exp(1) + 9))^(1/3))^n * ((3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2^(2/3))^(3/2) * sqrt(2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2)). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n+2*k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

A370875 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^2)) ).

Original entry on oeis.org

1, 1, 4, 24, 228, 2820, 44400, 840000, 18669840, 475871760, 13698296640, 439402803840, 15545690233920, 601352177025600, 25251437978807040, 1143932660001331200, 55612090342967558400, 2887929114414030086400, 159548423949650274739200
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2024

Keywords

Crossrefs

Programs

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

Formula

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

A371040 E.g.f. satisfies A(x) = exp(x^3*A(x)^2) / (1-x).

Original entry on oeis.org

1, 1, 2, 12, 96, 840, 9720, 143640, 2399040, 45239040, 976752000, 23537606400, 621444700800, 17936155036800, 562855739846400, 19038932398886400, 690456599575142400, 26748823900403404800, 1102407824344284057600, 48147134965603914240000
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sqrt(lambertw(-2*x^3/(1-x)^2)/(-2*x^3))))
    
  • PARI
    a(n) = n!*sum(k=0, n\3, (2*k+1)^(k-1)*binomial(n-k, n-3*k)/k!);

Formula

E.g.f.: sqrt(LambertW( -2*x^3/(1-x)^2 ) / (-2*x^3)).
a(n) = n! * Sum_{k=0..floor(n/3)} (2*k+1)^(k-1) * binomial(n-k,n-3*k)/k!.
Showing 1-4 of 4 results.