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.

A387273 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 16, 165, 1400, 10661, 75936, 517524, 3420960, 22123530, 140782048, 885008839, 5511579528, 34073731965, 209428887360, 1281220578936, 7808422173120, 47440778110398, 287490594872160, 1738463164498410, 10493677382085744, 63245915436539682, 380697445274657984
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A344055, A387272, A387274.
Cf. A387239, A387276.

Programs

  • Magma
    [&+[2^(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[2^(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, 2^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));

Formula

n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 12*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*x), with offset 3.

A387274 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).

Original entry on oeis.org

1, 20, 246, 2408, 20636, 162288, 1203000, 8546208, 58823919, 395245708, 2606333730, 16933021560, 108703640136, 691068080928, 4358220121296, 27301946599872, 170074452183570, 1054434358722024, 6510869338671852, 40063301434583504, 245781459952640040
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A344055, A387272, A387273.

Programs

  • Magma
    [&+[2^(n-k) * Binomial(n+4,k+4) * Binomial(2*k+8,k+8): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[2^(n-k)*Binomial[n+4,k+4]*Binomial[2*k+8,k+8],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(n+4, k+4)*binomial(2*k+8, k+8));

Formula

n*(n+8)*a(n) = (n+4) * (4*(2*n+7)*a(n-1) - 12*(n+3)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+4,n-2*k) * binomial(2*k+4,k).
a(n) = [x^n] (1+4*x+x^2)^(n+4).
E.g.f.: exp(4*x) * BesselI(4, 2*x), with offset 4.

A387272 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 12, 100, 720, 4815, 30884, 193144, 1188576, 7236690, 43741720, 263056728, 1576298464, 9421080123, 56200937940, 334801389360, 1992471776448, 11848869296622, 70425535830696, 418426332826200, 2485390365370080, 14760336569524854, 87650482093915752
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A344055, A387273, A387274.
Cf. A387280.

Programs

  • Magma
    [&+[2^(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[2^(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, 2^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));

Formula

n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 12*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+4*x+x^2)^(n+2).
E.g.f.: exp(4*x) * BesselI(2, 2*x), with offset 2.

A387278 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 10, 78, 560, 3885, 26550, 180285, 1221400, 8272251, 56062550, 380361212, 2583867720, 17575724491, 119705522370, 816297170310, 5572945684800, 38088275031435, 260576833989150, 1784382167211378, 12229792774162800, 83888652677196591, 575858959975595010
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A098409, A387275, A387276, A387277.
Cf. A026376, A344055.

Programs

  • Magma
    [&+[3^(n-k) * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025
  • Mathematica
    Table[Sum[3^(n-k)*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(n+1, k+1)*binomial(2*k+2, k+2));

Formula

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

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