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.

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.

A387238 Expansion of 1/((1-x) * (1-5*x))^(7/2).

Original entry on oeis.org

1, 21, 266, 2646, 22806, 178794, 1310694, 9140274, 61330269, 399107709, 2533330800, 15751925280, 96257031780, 579556206180, 3445117599480, 20252115155160, 117890464642335, 680320688005035, 3895668955041710, 22152779612619810, 125183331416173030
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A026375, A385563, A387237.
Cf. A126190, A374509, A387239.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-5*x))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-5*x))^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(7/2))

Formula

n*a(n) = (6*n+15)*a(n-1) - 5*(n+5)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387239(n).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).
Showing 1-3 of 3 results.

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