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.

A361816 Expansion of 1/sqrt(1 - 4*x*(1-x)^3).

Original entry on oeis.org

1, 2, 0, -10, -22, 12, 174, 344, -354, -3304, -5780, 9180, 65258, 99132, -226620, -1313580, -1690990, 5441340, 26681700, 28070100, -128211552, -543818824, -440381780, 2978145240, 11080939914, 6162798092, -68377892976, -225107280388, -64286124152
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2023

Keywords

Links

Crossrefs

Column k=3 of A361834.
Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361817.
Cf. A361812.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^3))

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(3*k,n-k).
n*a(n) = 2 * ( (2*n-1)*a(n-1) - 3*(2*n-2)*a(n-2) + 3*(2*n-3)*a(n-3) - (2*n-4)*a(n-4) ) for n > 3.

A361835 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 2, 2, -10, -10, 242, -678, -7054, 88342, -207646, -6015904, 88310862, -312514816, -8847633338, 184252541514, -1269592841970, -17662739133178, 634109114537218, -7914500471718552, -18165019012117450, 2936604063787679650, -62899139815867627378
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Links

Crossrefs

Main diagonal of A361834.
Cf. A361829, A361836.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k, k)*binomial(n*k, n-k));

Formula

a(n) = [x^n] 1/sqrt(1 - 4*x*(1-x)^n).

A361840 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 - x)^k)^(1/3).

Original entry on oeis.org

1, 1, 3, 1, 3, 18, 1, 3, 15, 126, 1, 3, 12, 90, 945, 1, 3, 9, 57, 585, 7371, 1, 3, 6, 27, 297, 3969, 58968, 1, 3, 3, 0, 78, 1629, 27657, 480168, 1, 3, 0, -24, -75, 207, 9216, 196290, 3961386, 1, 3, -3, -45, -165, -438, 459, 53217, 1411965, 33011550
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Examples

			Square array begins:
     1,    1,    1,   1,    1,    1, ...
     3,    3,    3,   3,    3,    3, ...
    18,   15,   12,   9,    6,    3, ...
   126,   90,   57,  27,    0,  -24, ...
   945,  585,  297,  78,  -75, -165, ...
  7371, 3969, 1629, 207, -438, -444, ...

Crossrefs

Columns k=0..3 give A004987, A361843, A361844, A361845.
Main diagonal gives A361847.
Cf. A361834, A361839.

Programs

  • PARI
    T(n, k) = (-1)^n*sum(j=0, n, 9^j*binomial(-1/3, j)*binomial(k*j, n-j));

Formula

n*T(n,k) = 3 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.
T(n,k) = (-1)^n * Sum_{j=0..n} 9^j * binomial(-1/3,j) * binomial(k*j,n-j).

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2*j,j) * binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 10, 32, 70, 1, 2, 12, 46, 136, 252, 1, 2, 14, 62, 226, 592, 924, 1, 2, 16, 80, 342, 1136, 2624, 3432, 1, 2, 18, 100, 486, 1932, 5810, 11776, 12870, 1, 2, 20, 122, 660, 3030, 11094, 30080, 53344, 48620
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Examples

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    2,   2,    2,    2,    2,    2, ...
    6,   8,   10,   12,   14,   16, ...
   20,  32,   46,   62,   80,  100, ...
   70, 136,  226,  342,  486,  660, ...
  252, 592, 1136, 1932, 3030, 4482, ...

Crossrefs

Columns k=0..5 give A000984, A006139, A137635, A361812, A361813, A361814.
Main diagonal gives A361829.
Cf. A099233, A361834.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(2*j, j)*binomial(k*j, n-j));

Formula

G.f. of column k: 1/sqrt(1 - 4*x*(1+x)^k).
n*T(n,k) = 2 * Sum_{j=0..k} binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.
Showing 1-4 of 4 results.

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