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.

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.

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.

A387281 Expansion of 1/((1-2*x) * (1-6*x))^(7/2).

Original entry on oeis.org

1, 28, 462, 5880, 63966, 626472, 5692764, 48919728, 402648246, 3202791592, 24780247492, 187393703952, 1390208264172, 10146829592592, 73029572999352, 519260074512480, 3652939914500646, 25457292175929768, 175932472247239092, 1206772898939860560, 8221969006750158660
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A387273.

Programs

  • Mathematica
    Module[{x}, CoefficientList[Series[1/((3*x - 2)*4*x + 1)^(7/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(7/2))

Formula

n*a(n) = (8*n+20)*a(n-1) - 12*(n+5)*a(n-2) for n > 1.
a(n) = (-2)^n * Sum_{k=0..n} 3^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = 2^n * Sum_{k=0..n} (-2)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 4^k * 6^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387273(n).
a(n) = (-1)^n * Sum_{k=0..n} 8^k * (3/2)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).
Showing 1-4 of 4 results.

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