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.

A361845 Expansion of 1/(1 - 9*x*(1-x)^3)^(1/3).

Original entry on oeis.org

1, 3, 9, 27, 78, 207, 459, 567, -1926, -20763, -120123, -569349, -2410200, -9379449, -33818715, -112292001, -335018295, -837341388, -1317232530, 2358000072, 35974607355, 228270292803, 1148026536963, 5094839173779, 20667058966044, 77501033284779
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Links

Crossrefs

Column k=3 of A361840.
Cf. A361816.

Programs

  • Mathematica
    a[n_]:=(-9)^n*Binomial[-1/3, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 2/3-n, 2/3-n}, 2^8/3^5]; Array[a,26,0] (* Stefano Spezia, Jul 11 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1-x)^3)^(1/3))

Formula

n*a(n) = 3 * ( (3*n-2)*a(n-1) - 3*(3*n-4)*a(n-2) + 3*(3*n-6)*a(n-3) - (3*n-8)*a(n-4) ) for n > 3.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(3*k,n-k).
a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 2/3-n, 2/3-n], 2^8/3^5). - Stefano Spezia, Jul 11 2024

A361815 Expansion of 1/sqrt(1 - 4*x*(1-x)^2).

Original entry on oeis.org

1, 2, 2, -2, -14, -32, -30, 64, 346, 752, 584, -2044, -9486, -19324, -11368, 66180, 271658, 514916, 192584, -2151612, -7949736, -13933280, -1779028, 69933368, 235295106, 378579404, -61171228, -2267724644, -7003832456, -10248117752, 5236354188, 73288104568
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2023

Keywords

Comments

Diagonal of rational function 1/(1 - (1 - x*y) * (x + y)).

Links

Crossrefs

Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361816, A361817.
Cf. A137635.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^2))

Formula

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

A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2*j,j) * binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 4, 20, 1, 2, 2, 8, 70, 1, 2, 0, -2, 16, 252, 1, 2, -2, -10, -14, 32, 924, 1, 2, -4, -16, -22, -32, 64, 3432, 1, 2, -6, -20, -10, 12, -30, 128, 12870, 1, 2, -8, -22, 20, 118, 174, 64, 256, 48620, 1, 2, -10, -22, 66, 242, 304, 344, 346, 512, 184756
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Examples

			Square array begins:
    1,  1,   1,   1,   1,   1,    1, ...
    2,  2,   2,   2,   2,   2,    2, ...
    6,  4,   2,   0,  -2,  -4,   -6, ...
   20,  8,  -2, -10, -16, -20,  -22, ...
   70, 16, -14, -22, -10,  20,   66, ...
  252, 32, -32,  12, 118, 242,  342, ...
  924, 64, -30, 174, 304,  82, -678, ...

Crossrefs

Columns k=0..4 give A000984, A000079, A361815, A361816, A361817.
Main diagonal gives A361835.
Cf. A361830.

Programs

  • PARI
    T(n, k) = sum(j=0, n, (-1)^(n-j)*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} (-1)^j * binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.

A361817 Expansion of 1/sqrt(1 - 4*x*(1-x)^4).

Original entry on oeis.org

1, 2, -2, -16, -10, 118, 304, -500, -3754, -2488, 30866, 83716, -135568, -1080972, -792876, 9090484, 25788118, -39325156, -335074520, -271779024, 2820643842, 8348113120, -11788972644, -107836934448, -96107852032, 900943403012, 2778574561276, -3596374190416
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2023

Keywords

Crossrefs

Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361816.
Cf. A361813.

Programs

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

Formula

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

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