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

A331515 Expansion of 1/(1 - 8*x + 4*x^2)^(3/2).

Original entry on oeis.org

1, 12, 114, 1000, 8430, 69384, 561988, 4499856, 35719830, 281634760, 2208564732, 17242680624, 134118558028, 1039939550160, 8041848166920, 62042202765856, 477670318108902, 3670988584476744, 28166853684793420, 215807899372086000, 1651323989374972836
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Links

Crossrefs

Column 4 of A331514.
Cf. A007564, A069835, A385728.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 21); Coefficients(R!( 1/(1 - 8*x + 4*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020
  • Magma
    [&+[2^(n-k)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..21]]; // Marius A. Burtea, Jan 20 2020
  • Mathematica
    a[n_] := Sum[2^(n - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n + 1}]; Array[a, 21, 0] (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(3/2))
  • PARI
    a(n) = sum(k=1, n+1, 2^(n-k)*k*binomial(n+1, k)*binomial(n+1+k, k));

Formula

a(n) = Sum_{k=1..n+1} 2^(n-k) * k * binomial(n+1,k) * binomial(n+1+k,k).
n * a(n) = 4 * (2*n+1) * a(n-1) - 4 * (n+1) * a(n-2) for n>1.
a(n) = ((n+2)/2) * Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).
a(n) ~ 2^(n - 1/2) * (2 + sqrt(3))^(n + 3/2) * sqrt(n) / (3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = binomial(n+2,2) * A007564(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} 2^k * (-1/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)

A385813 Expansion of 1/((1-3*x) * (1-7*x))^(3/2).

Original entry on oeis.org

1, 15, 156, 1400, 11655, 92925, 721140, 5496300, 41361255, 308344025, 2282167272, 16795140180, 123030071437, 897791417775, 6530377362480, 47370038320800, 342794475282915, 2475479922896925, 17843821672113780, 128412824128709400, 922775179449162501, 6622378039719342615
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A385563, A385728.
Cf. A098409, A182401.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 25 2025
  • Mathematica
    CoefficientList[Series[1/((1-3x)*(1-7*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 25 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(3/2))

Formula

n*a(n) = (10*n+5)*a(n-1) - 21*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A182401(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (5/2)^k * (-21/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
Showing 1-3 of 3 results.

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