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 1-4 of 4 results.

A387338 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k) * binomial(n+3,k+3).

Original entry on oeis.org

1, 12, 100, 720, 4809, 30744, 191184, 1167120, 7033785, 41999364, 249075684, 1469561184, 8636441905, 50600529840, 295755641152, 1725379046496, 10050215851665, 58470232877820, 339832224226180, 1973538115293360, 11453616812552761, 66436765880135112
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2025

Keywords

Links

Crossrefs

Cf. A002696, A387339.
Cf. A387342.

Programs

  • Magma
    [&+[2^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
  • Mathematica
    Table[Sum[2^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(n+3, k)*binomial(n+3, k+3));

Formula

n*(n+6)*a(n) = (n+3) * (3*(2*n+5)*a(n-1) - (n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^k * 3^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+3*x+2*x^2)^(n+3).
E.g.f.: exp(3*x) * BesselI(3, 2*sqrt(2)*x) / (2*sqrt(2)), with offset 3.

A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

Original entry on oeis.org

1, 16, 175, 1640, 14189, 117152, 939036, 7379040, 57188010, 438810592, 3342302821, 25316084248, 190937278805, 1435287936320, 10760879892008, 80509920297792, 601343784616830, 4485466826475360, 33420579148668670, 248788060638391120, 1850652536242372786
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2025

Keywords

Links

Crossrefs

Cf. A069835, A331792, A387339.
Cf. A002696, A387338.

Programs

  • Magma
    [&+[3^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
  • Mathematica
    Table[Sum[3^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(n+3, k)*binomial(n+3, k+3));

Formula

n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 4*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*sqrt(3)*x) / (3*sqrt(3)), with offset 3.

A387369 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+2,k) * binomial(n+2,n-k).

Original entry on oeis.org

1, 15, 174, 1850, 18915, 189525, 1877596, 18476820, 181083285, 1770245675, 17278828842, 168496597230, 1642259489143, 16002398658225, 155919866646840, 1519307275471400, 14806582620440553, 144329229195062535, 1407215890063071910, 13724133021646678050, 133885448856624266571
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2025

Keywords

Links

Crossrefs

Cf. A006442, A387368.
Cf. A387339.

Programs

  • Magma
    [&+[2^k * 3^(n-k) * Binomial(n+2,k) * Binomial(n+2,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
  • Mathematica
    Table[Sum[2^k * 3^(n-k)*Binomial[n+2,k]*Binomial[n+2, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(n+2, k)*binomial(n+2, n-k));

Formula

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+2,k) * binomial(n+2,n-k).
n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - (n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 6^k * 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+6*x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 2*sqrt(6)*x) / 6, with offset 2.

A387343 Expansion of 1/(1 - 8*x + 4*x^2)^(5/2).

Original entry on oeis.org

1, 20, 270, 3080, 31990, 312984, 2937900, 26751120, 237977190, 2078447800, 17884238372, 152002796400, 1278603975740, 10660760170480, 88213513627800, 725107271106336, 5925674432448390, 48175954959638520, 389871795632108020, 3142078444590396080, 25228464363569709396
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2025

Keywords

Links

Crossrefs

Cf. A069835, A331515, A387344.
Cf. A002802, A387313, A387341.
Cf. A387339.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 8*x + 4*x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/(1-8*x+4*x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(5/2))

Formula

n*a(n) = 4*(2*n+3)*a(n-1) - 4*(n+3)*a(n-2) for n > 1.
a(n) = (binomial(n+4,2)/6) * A387339(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/2)^(n-4*k) * binomial(-5/2,k) * binomial(k,n-k).
Showing 1-4 of 4 results.

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