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 1-3 of 3 results.

A370928 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x^3)) ).

Original entry on oeis.org

1, 1, 4, 30, 360, 5880, 120960, 2996280, 86889600, 2889976320, 108501724800, 4539844108800, 209497816281600, 10570762445443200, 578997352591257600, 34214810278128480000, 2169772724008976486400, 146984464202544531763200
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Crossrefs

Programs

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

Formula

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

A370984 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2*exp(x)) ).

Original entry on oeis.org

1, 0, 2, 6, 84, 860, 14430, 257082, 5678456, 140241096, 3952791450, 123539438990, 4266378769092, 160943793753756, 6592371152535350, 291260465060881890, 13809548247503299440, 699362685890810753552, 37679514498664685654706
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Crossrefs

Programs

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

Formula

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

A371067 E.g.f. satisfies A(x) = 1 + x*exp(x^2*A(x)^2).

Original entry on oeis.org

1, 1, 0, 6, 48, 180, 2880, 46200, 483840, 9087120, 203212800, 3752511840, 89413632000, 2510276408640, 66301996400640, 1982685238934400, 67064515854336000, 2274167610024710400, 82881756045036748800, 3301346557970183923200, 135363022243685203968000
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(2*k+1,n-2*k)/( (2*k+1)*k! ).
a(n) ~ n^(n-1) / (sqrt(2) * exp(n) * r^(n+1)), where r = 0.450347181930267755599214125867779338412791581819135528888185619948594... and s = 2.1478259175343697310213089706837271102656629945040966643073615920885... are roots of the system of equations exp(r^2*s^2)*r = s-1, 2*(s-1)*r^2*s = 1. - Vaclav Kotesovec, Mar 10 2024
Showing 1-3 of 3 results.