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.

A387229 Expansion of sqrt((1-x) / (1-9*x)^5).

Original entry on oeis.org

1, 22, 343, 4604, 56765, 662450, 7438515, 81174840, 866564025, 9090485390, 94014360143, 960890353076, 9723664642549, 97564323687082, 971756818248235, 9616894723897200, 94635806917660785, 926607762721058310, 9032093873432341575, 87685949210949054060, 848182216775168898861
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A387228, A387230.
Cf. A085363, A387208.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-9*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025
  • Mathematica
    CoefficientList[Series[Sqrt[(1-x)/(1-9*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-9*x)^5))

Formula

n*a(n) = (10*n+12)*a(n-1) - 9*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 2^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 2^(7/2) * n^(3/2) * 3^(2*n-2) / sqrt(Pi). - Vaclav Kotesovec, Aug 24 2025

A387230 Expansion of sqrt((1-x) / (1-13*x)^5).

Original entry on oeis.org

1, 32, 723, 14044, 250415, 4224732, 68565049, 1081299296, 16679767923, 252819395920, 3777709472537, 55782986878164, 815526073468561, 11821376147023268, 170096339292264375, 2431786467331116016, 34569517907583692867, 488963045591838160848, 6885041951078984405449
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A387228, A387229.
Cf. A085364, A387210.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-13*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025
  • Mathematica
    CoefficientList[Series[Sqrt[(1-x)/(1-13*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sqrt((1-x)/(1-13*x)^5))

Formula

n*a(n) = (14*n+18)*a(n-1) - 13*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 3^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * n^(3/2) * 13^(n - 1/2) / sqrt(3*Pi). - Vaclav Kotesovec, Aug 24 2025

A387233 Expansion of sqrt((1-2*x) / (1-6*x)^5).

Original entry on oeis.org

1, 14, 142, 1252, 10190, 78724, 586236, 4247688, 30132438, 210175540, 1445920388, 9833940472, 66237449356, 442463439656, 2934485313400, 19340115356688, 126759642351462, 826734451831956, 5368338057048756, 34721155684000920, 223765535492622564, 1437403425873718776
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A387228, A387234.
Cf. A360317, A387211.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 2*x) / (1-6*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025
  • Mathematica
    CoefficientList[Series[Sqrt[(1-2*x)/(1-6*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)^5))

Formula

n*a(n) = (8*n+6)*a(n-1) - 12*n*a(n-2) for n > 1.
a(n) = (1/2)^n * Sum_{k=0..n} 3^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 2^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).

A387234 Expansion of sqrt((1-3*x) / (1-7*x)^5).

Original entry on oeis.org

1, 16, 187, 1908, 18015, 161700, 1400385, 11808480, 97533075, 792374720, 6350977457, 50334074972, 395137260609, 3076728075036, 23787996024015, 182783869074000, 1396834725138435, 10622886492055680, 80436297856668225, 606683298398776620, 4559675718517366461
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A387228, A387233.
Cf. A360318, A387212.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 3*x) / (1-7*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025
  • Mathematica
    CoefficientList[Series[Sqrt[(1-3*x)/(1-7*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)^5))

Formula

n*a(n) = (10*n+6)*a(n-1) - 21*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 3^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
Showing 1-4 of 4 results.

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