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.

A369215 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x) ).

Original entry on oeis.org

1, 4, 29, 261, 2627, 28315, 319648, 3731037, 44663058, 545312504, 6764556591, 85015779095, 1080185111768, 13852183882612, 179058158369828, 2330621446075640, 30519758687849439, 401806204894374041, 5315243189757111099, 70613088335938995385, 941714812929017751855
Offset: 0

Views

Author

Seiichi Manyama, Jan 16 2024

Keywords

Links

Crossrefs

Cf. A369114, A369161.
Cf. A151374, A249924, A369216.
Cf. A365764, A379171, A379172.
Cf. A049140.

Programs

  • Mathematica
    CoefficientList[InverseSeries[Series[x((1-x)^3-x),{x,0,21}],x]/x,x] (* Stefano Spezia, Mar 31 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x))/x)
  • PARI
    a(n) = sum(k=0, n, binomial(n+k, k)*binomial(4*n+2*k+2, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(4*n+2*k+2,n-k).

A379172 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3)/(1 - x*A(x))^3.

Original entry on oeis.org

1, 4, 33, 358, 4445, 59745, 846023, 12430941, 187753479, 2896929975, 45465112431, 723520554096, 11647721390271, 189352106241567, 3104046096391902, 51254005259550753, 851674902290491936, 14231191062537888864, 238978853442142491358, 4030889937027642017872
Offset: 0

Views

Author

Seiichi Manyama, Dec 17 2024

Keywords

Crossrefs

Cf. A365764, A369215, A379171.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(4*n+5*k+2, n-k)/(n+2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(4*n+5*k+2,n-k)/(n+2*k+1).

A379173 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^2.

Original entry on oeis.org

1, 3, 11, 53, 284, 1630, 9794, 60830, 387390, 2515892, 16599051, 110943779, 749603067, 5111606801, 35133394554, 243146923574, 1692918638012, 11850006727400, 83341778073920, 588646472669454, 4173607638548291, 29694593381322531, 211941668053441490, 1517087043428034420
Offset: 0

Views

Author

Seiichi Manyama, Dec 17 2024

Keywords

Crossrefs

Cf. A003169, A249924, A379174.
Cf. A025227, A379171.

Programs

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

Formula

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