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.

A379025 a(n) = Sum_{k=0..n} binomial(3*n+k-1,k) * binomial(3*n+k,n-k).

Original entry on oeis.org

1, 6, 78, 1149, 17850, 285711, 4661727, 77086008, 1287322866, 21661521945, 366687839133, 6237631866417, 106535632157643, 1825763898882189, 31379978657609100, 540688387589377764, 9336602657251874754, 161534120354250452361, 2799488717098336992687
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Links

Crossrefs

Cf. A262910, A379022, A379026.

Programs

  • Magma
    [&+[Binomial(3*n+k-1, k)*Binomial(3*n+k, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Dec 21 2024
  • Mathematica
    Table[Sum[Binomial[3*n+k-1,k]*Binomial[3*n+k, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Dec 21 2024 *)
a[n_]:= Binomial[3*n, n]*HypergeometricPFQ[{-n, 3*n, 1 + 3*n}, {1/2 + n, 1 + n}, -1/4]; Array[a,19,0] (* Stefano Spezia, Dec 22 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*n+k-1, k)*binomial(3*n+k, n-k));

Formula

a(n) = [x^n] ( (1 + x)/(1 - x - x^2) )^(3*n).
a(n) = binomial(3*n, n)*hypergeom([-n, 3*n, 1 + 3*n], [1/2 + n, 1 + n], -1/4). - Stefano Spezia, Dec 22 2024

A379022 a(n) = Sum_{k=0..n} binomial(2*n+k-1,k) * binomial(2*n+k,n-k).

Original entry on oeis.org

1, 4, 36, 370, 4012, 44824, 510498, 5892310, 68684540, 806715964, 9532070396, 113179713046, 1349276883346, 16140148109960, 193629588953214, 2328744593780590, 28068490664161756, 338960821947139640, 4100329281075440400, 49676100591186493156, 602654837914634224812
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Crossrefs

Cf. A262910, A379025, A379026.
Cf. A379021.

Programs

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

Formula

a(n) = [x^n] ( (1 + x)/(1 - x - x^2) )^(2*n).

A379024 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^4 ).

Original entry on oeis.org

1, 8, 100, 1500, 24846, 438064, 8062518, 153117320, 2978260865, 59031215508, 1187987779084, 24210092837648, 498606095949315, 10361291534825800, 216982960825089730, 4574651332139656108, 97018731642209493810, 2068350691029593934000, 44301394943232879298360
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Links

Crossrefs

Cf. A007863, A379021, A379023.
Cf. A239108, A379026.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379026(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(5/4)) )^4.
(3) A(x) = B(x)^4 where B(x) is the g.f. of A239108.
a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(4*(n+1)).
a(n) = 4 * Sum_{k=0..n} binomial(4*n+k+4,k) * binomial(4*n+k+4,n-k)/(4*n+k+4) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(4*n+k+4,n-k).
Showing 1-3 of 3 results.

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

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