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.

A365121 G.f. A(x) satisfies A(x) = (1 + x / (1 - x*A(x))^2)^3.

Original entry on oeis.org

1, 3, 9, 40, 192, 993, 5375, 30081, 172650, 1010640, 6010530, 36214656, 220590082, 1356131892, 8403647454, 52436122717, 329170499604, 2077465903503, 13173914483799, 83897445169341, 536355204428412, 3440875097256529, 22144300030907667
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2023

Keywords

Crossrefs

Cf. A000108, A365120.
Cf. A365119, A365122.
Cf. A367242.

Programs

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

Formula

If g.f. satisfies A(x) = (1 + x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*(n-k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A367242. - Seiichi Manyama, Dec 06 2024

A371616 G.f. satisfies A(x) = 1 + x / (1 - x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 3, 15, 82, 495, 3147, 20812, 141621, 985287, 6976369, 50108232, 364202415, 2673756449, 19797659586, 147677816532, 1108711280376, 8371222635096, 63525564996093, 484243596619753, 3706268752629237, 28470703720193010, 219432896755734137
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2024

Keywords

Links

Crossrefs

Cf. A364742, A367242.
Cf. A001764, A365113.
Cf. A002293, A367281.

Programs

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

Formula

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

A367280 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 5, 33, 251, 2073, 18069, 163600, 1523731, 14504988, 140499307, 1380322749, 13721269995, 137758098052, 1394840743638, 14227181658075, 146048314214619, 1507739540085350, 15643456882376418, 163036276218805231, 1706021256401103673
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2023

Keywords

Crossrefs

Cf. A002294, A366176, A367232, A367238.
Cf. A002293, A367242.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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