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.

A387276 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 20, 255, 2650, 24521, 210840, 1725234, 13631700, 104993955, 793367300, 5907885412, 43495473840, 317355930255, 2298888740400, 16555878011448, 118661449810320, 847132614218907, 6027874235210700, 42773816956415055, 302816249208061050, 2139537520524710691
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A387275, A387277, A387278.
Cf. A387239, A387273.

Programs

  • Magma
    [&+[3^(n-k) * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[3^(n-k)*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));

Formula

n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 21*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+5*x+x^2)^(n+3).
E.g.f.: exp(5*x) * BesselI(3, 2*x), with offset 3.

A387275 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 15, 154, 1350, 10890, 83650, 623056, 4547520, 32735085, 233369675, 1652203542, 11638252730, 81674873553, 571575363975, 3991529920440, 27829484027400, 193791573179883, 1348196149698885, 9372495529924710, 65120144658997050, 452263192928596896
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A387276, A387277, A387278.

Programs

  • Magma
    [&+[3^(n-k) * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[3^(n-k)*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));

Formula

n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 21*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 2*x), with offset 2.
a(n) ~ 7^(n + 5/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 31 2025

A387277 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).

Original entry on oeis.org

1, 25, 381, 4585, 47978, 458010, 4100370, 35027850, 288845370, 2317794050, 18203687502, 140533725150, 1069904389008, 8052575725680, 60033791987424, 444015014417280, 3261950250436845, 23827019766988725, 173193081555808545, 1253583401573658925, 9040278899072328006
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A387275, A387276, A387278.
Cf. A387274.

Programs

  • Magma
    [&+[3^(n-k) * Binomial(n+4,k+4) * Binomial(2*k+8,k+8): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025
  • Mathematica
    Table[Sum[3^(n-k)*Binomial[n+4,k+4]*Binomial[2*k+8,k+8],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(n+4, k+4)*binomial(2*k+8, k+8));

Formula

n*(n+8)*a(n) = (n+4) * (5*(2*n+7)*a(n-1) - 21*(n+3)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+4,n-2*k) * binomial(2*k+4,k).
a(n) = [x^n] (1+5*x+x^2)^(n+4).
E.g.f.: exp(5*x) * BesselI(4, 2*x), with offset 4.
Showing 1-3 of 3 results.

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