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.

A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2*k*x + ((k-2)*x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 3, -6, 1, 6, 6, 0, 1, 9, 30, 10, 30, 1, 12, 66, 140, 15, 0, 1, 15, 114, 450, 630, 21, -140, 1, 18, 174, 1000, 2955, 2772, 28, 0, 1, 21, 246, 1850, 8430, 18963, 12012, 36, 630, 1, 24, 330, 3060, 18855, 69384, 119812, 51480, 45, 0
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Examples

			Square array begins:
    1,  1,    1,     1,     1,      1, ...
    0,  3,    6,     9,    12,     15, ...
   -6,  6,   30,    66,   114,    174, ...
    0, 10,  140,   450,  1000,   1850, ...
   30, 15,  630,  2955,  8430,  18855, ...
    0, 21, 2772, 18963, 69384, 187425, ...

Links

Crossrefs

Columns k=1..5 give A000217(n+1), A002457, A002695(n+1), A331515, A331516.
Cf. A307883, A331511.

Programs

  • Mathematica
    T[n_, k_] = 1/2 * Sum[If[k == 2 && n == j - 1, 1, (k - 2)^(n + 1 - j)] * j * Binomial[n + 1, j] * Binomial[n + 1 + j, j], {j, 1, n + 1}]; Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    T(n,k) = (1/2)*sum(j=1,n+1,(k-2)^(n+1-j)*j*binomial(n+1,j)*binomial(n+1+j,j));
matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jan 20 2020

Formula

T(n,k) = (1/2) * Sum_{j=1..n+1} (k-2)^(n+1-j) * j * binomial(n+1,j) * binomial(n+1+j,j).
n * T(n,k) = k * (2*n+1) * T(n-1,k) - (k-2)^2 * (n+1) * T(n-2,k) for n > 1.
T(n,k) = ((n+2)/2) * Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
T(n,k) = Sum_{j=0..n} (k/2)^j * (-(k-2)^2/(2*k))^(n-j) * (2*j+1) * binomial(2*j,j) * binomial(j,n-j) for k > 0. - Seiichi Manyama, Aug 20 2025

A331792 Expansion of ((1 - 4*x)/sqrt(1 - 8*x + 4*x^2) - 1)/(6*x^2).

Original entry on oeis.org

1, 8, 57, 400, 2810, 19824, 140497, 999968, 7143966, 51206320, 368094122, 2652720096, 19159794004, 138658606688, 1005231020865, 7299082678336, 53074479789878, 386419850997552, 2816685368479342, 20553133273532000, 150120362670452076
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2020

Keywords

Links

Crossrefs

Column 4 of A331791.
Cf. A069835, A331515.

Programs

  • Mathematica
    a[n_] := Sum[3^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, May 05 2021 *)
  • PARI
    N=20; x='x+O('x^N); Vec(((1-4*x)/sqrt(1-8*x+4*x^2)-1)/(6*x^2))
  • PARI
    {a(n) = sum(k=0, n, 3^k*binomial(n+1, k)*binomial(n+1, k+1))}

Formula

a(n) = (2/(n+2)) * A331515(n) = Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).
n * (n+2) * a(n) = (n+1) * (4 * (2*n+1) * a(n-1) - 4 * n * a(n-2)) for n>1.
a(n) ~ 2^(n + 1/2) * (2 + sqrt(3))^(n + 3/2) / (3^(3/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 26 2020
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - Seiichi Manyama, Aug 24 2025
From Seiichi Manyama, Aug 27 2025: (Start)
a(n) = [x^n] (1+4*x+3*x^2)^(n+1).
E.g.f.: exp(4*x) * BesselI(1, 2*sqrt(3)*x) / sqrt(3), with offset 1. (End)

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

A387343 Expansion of 1/(1 - 8*x + 4*x^2)^(5/2).

Original entry on oeis.org

1, 20, 270, 3080, 31990, 312984, 2937900, 26751120, 237977190, 2078447800, 17884238372, 152002796400, 1278603975740, 10660760170480, 88213513627800, 725107271106336, 5925674432448390, 48175954959638520, 389871795632108020, 3142078444590396080, 25228464363569709396
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2025

Keywords

Links

Crossrefs

Cf. A069835, A331515, A387344.
Cf. A002802, A387313, A387341.
Cf. A387339.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 8*x + 4*x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/(1-8*x+4*x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(5/2))

Formula

n*a(n) = 4*(2*n+3)*a(n-1) - 4*(n+3)*a(n-2) for n > 1.
a(n) = (binomial(n+4,2)/6) * A387339(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/2)^(n-4*k) * binomial(-5/2,k) * binomial(k,n-k).
Showing 1-4 of 4 results.

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