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.

A385563 Expansion of 1/((1-x) * (1-5*x))^(3/2).

Original entry on oeis.org

1, 9, 60, 360, 2055, 11403, 62132, 334260, 1781415, 9425295, 49581576, 259601004, 1353939405, 7038232425, 36484340400, 188665670880, 973545780195, 5014258620075, 25783103206100, 132378800689800, 678768332410245, 3476164133573505, 17782899991147500
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Partial sums of A383254.
Cf. A331516, A385716.
Cf. A385728, A385813.
Cf. A002212, A383949.

Programs

  • Mathematica
    Module[{a, n}, RecurrenceTable[{a[n] == ((6*n+3)*a[n-1] - 5*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 9}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(3/2))

Formula

n*a(n) = (6*n+3)*a(n-1) - 5*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A002212(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (3/2)^k * (-5/6)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(n) * 5^(n + 3/2) / (4*sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

A385728 Expansion of 1/((1-2*x) * (1-6*x))^(3/2).

Original entry on oeis.org

1, 12, 102, 760, 5310, 35784, 235788, 1530288, 9824310, 62557000, 395797908, 2491381776, 15616141996, 97537784400, 607391245080, 3772617319008, 23379854507046, 144605546475336, 892834113930180, 5504041611527760, 33883431379007364, 208327771987901808
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A385563, A385813.
Cf. A005572, A081671, A331515.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1 / ((1 - 2*x) * (1 - 6*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 22 2025
  • Mathematica
    Module[{a, n}, RecurrenceTable[{a[n] == ((8*n+4)*a[n-1] - 12*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 12}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)
CoefficientList[Series[ 1/((1-2*x)*(1-6*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 22 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(3/2))

Formula

n*a(n) = (8*n+4)*a(n-1) - 12*(n+1)*a(n-2) for n > 1.
a(n) = (1/2)^n * Sum_{k=0..n} 3^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A005572(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} 2^k * (-3/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(n) * 2^(n - 1/2) * 3^(n + 3/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 21 2025

A387212 Expansion of sqrt((1-3*x) / (1-7*x)^3).

Original entry on oeis.org

1, 9, 75, 599, 4659, 35595, 268485, 2005785, 14873715, 109643195, 804354417, 5877232773, 42798735805, 310767250773, 2250899498763, 16267896905895, 117347641620435, 845043416086635, 6076092412278465, 43629213402099045, 312892629725930121, 2241442380182752209
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2025

Keywords

Links

Crossrefs

Cf. A163869, A387211.
Cf. A360318, A385813.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 3*x) / (1-7*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025
  • Mathematica
    CoefficientList[Series[Sqrt[(1-3*x)/(1-7*x)^3],{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)^3))

Formula

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

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