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.

Previous Showing 11-13 of 13 results.

A336829 a(n) = Sum_{k=0..n} binomial(n+k,k)^n.

Original entry on oeis.org

1, 3, 46, 9065, 25561876, 1048567813062, 632156164654144530, 5652307059542612442465921, 755658094192422806457805924637704, 1521188219372604726826961340683399629967888, 46388428590466766659538640978460161019178279424832676
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 05 2020

Keywords

Links

Crossrefs

Cf. A001700, A112028, A112029, A167010, A219562, A219563, A219564.

Programs

  • Magma
    [(&+[Binomial(2*n-j,n)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022
  • Mathematica
    Table[Sum[Binomial[n + k, k]^n, {k, 0, n}], {n, 0, 10}]
  • PARI
    a(n) = sum(k=0, n, binomial(n+k, k)^n); \\ Michel Marcus, Aug 05 2020
  • SageMath
    def A336829(n): return sum(binomial(2*n-j, n)^n for j in (0..n))
[A336829(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

Formula

a(n) ~ exp(-1/8) * 2^(2*n^2) / (Pi*n)^(n/2). - Vaclav Kotesovec, Jul 10 2021

A374675 a(n) = Sum_{k=0..n} binomial(n+k-1,n) * binomial(n+k,n).

Original entry on oeis.org

1, 2, 21, 244, 3055, 40116, 544103, 7553960, 106734195, 1528937200, 22143771386, 323613807672, 4765050473521, 70611124679564, 1052084386116915, 15750153504353872, 236766112174722235, 3572264808332407512, 54073065371555283968, 820886646834780680640
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2024

Keywords

Crossrefs

Cf. A001791, A374676.
Cf. A112029, A129763, A259335.

Programs

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

Formula

a(n) = Sum_{k=0..n} (k/(n+k)) * binomial(n+k,k)^2 for n > 0.
a(n) = (n+1) * A259335(n).

A382403 a(n) = Sum_{k=0..n} A039599(n,k)^3.

Original entry on oeis.org

1, 2, 36, 980, 33040, 1268568, 53105976, 2364239592, 110206067400, 5323547715200, 264576141331216, 13458185494436592, 697931136204820336, 36789784967375728400, 1966572261077797609200, 106400946932857148590800, 5817987630644593688220600, 321105713814359742307398480
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2025

Keywords

Comments

Let b_k(n) = Sum_{j=0..n} A039599(n,j)^k. b_1(n) = binomial(2*n,n) = A000984(n) and b_2(n) = binomial(4*n,2*n)/(2*n+1) = A048990(n).

Links

Crossrefs

Cf. A000984, A048990.
Cf. A039599, A112029.

Programs

  • PARI
    a039599(n, k) = (2*k+1)/(n+k+1)*binomial(2*n, n-k);
a(n) = sum(k=0, n, a039599(n, k)^3);

Formula

a(n) = binomial(2*n,n) * (4 * binomial(2*n,n)^2 - 3 * A112029(n)).
Previous Showing 11-13 of 13 results.

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

Content is available under The OEIS End-User License Agreement.