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.

A365182 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 5, 33, 252, 2091, 18319, 166750, 1561599, 14948572, 145615404, 1438752770, 14384289530, 145248707646, 1479212551278, 15175516654760, 156691764630780, 1627069871618145, 16980373299730925, 178006989972532900, 1873607777794186000
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Programs

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

Formula

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

A365175 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)).

Original entry on oeis.org

1, 1, 10, 189, 5476, 215145, 10701006, 644909503, 45687408712, 3721382812305, 342689189598010, 35206864089944151, 3992473080042706524, 495361299387667990537, 66752437447119717428422, 9708649781691227748131535, 1515863453268825963300368656
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Programs

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

Formula

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

A365176 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 10, 195, 5836, 236925, 12177966, 758458603, 55528414264, 4674208189977, 444823048027450, 47227542351423951, 5534636939373353604, 709653811287800826421, 98825110036657191358822, 14853654178825132742729715, 2396666529204491489278153456
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Programs

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

Formula

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

A377548 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^3 ).

Original entry on oeis.org

1, 3, 36, 789, 25644, 1112655, 60584058, 3975599271, 305587795320, 26941234079259, 2680537845979470, 297158198268036963, 36325021999771692036, 4854553774172042934279, 704185171457954845825026, 110192472149320674192100815, 18503193203651913813111781488, 3318723221891108953801703239731
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 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) = 3*n!*sum(k=0, n, k^(n-k)*binomial(3*n+k+3, k)/((3*n+k+3)*(n-k)!));

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A365177.
a(n) = 3 * n! * Sum_{k=0..n} k^(n-k) * binomial(3*n+k+3,k)/( (3*n+k+3)*(n-k)! ).

A377631 E.g.f. satisfies A(x) = 1/(1 - x * A(x)^4 * exp(x*A(x)^4)).

Original entry on oeis.org

1, 1, 12, 297, 11380, 593785, 39304206, 3155996557, 298106913336, 32391139027185, 3980284376962330, 545806093612966021, 82628400115183659012, 13688201250584241332809, 2463065653446247669021398, 478399017659163635014545405, 99757368661138669886988396016
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*n+k+1,k)/( (4*n+k+1)*(n-k)! ).
Showing 1-5 of 5 results.