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.

A361839 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, 21, 126, 1, 3, 24, 162, 945, 1, 3, 27, 201, 1341, 7371, 1, 3, 30, 243, 1809, 11529, 58968, 1, 3, 33, 288, 2352, 16893, 101619, 480168, 1, 3, 36, 336, 2973, 23607, 161676, 911466, 3961386, 1, 3, 39, 387, 3675, 31818, 242757, 1574289, 8281737, 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,    21,    24,    27,    30,    33, ...
   126,   162,   201,   243,   288,   336, ...
   945,  1341,  1809,  2352,  2973,  3675, ...
  7371, 11529, 16893, 23607, 31818, 41676, ...

Crossrefs

Columns k=0..3 give A004987, A180400, A361841, A361842.
Main diagonal gives A361846.
Cf. A361830, A361840.

Programs

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

A361881 Expansion of 1/(1 - 9*x/(1 + x))^(1/3).

Original entry on oeis.org

1, 3, 15, 93, 618, 4278, 30390, 219810, 1611105, 11929395, 89045079, 669018837, 5053759440, 38350056072, 292147584072, 2233020788184, 17117923408746, 131560216858110, 1013413369611606, 7822237588031586, 60487791859818348, 468511159492134516
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

Cf. A004987, A180400, A361841, A361842, A361882.
Cf. A361375.

Programs

  • Maple
    a := n -> if n = 0 then 1 else (-1)^(1-n)*3*hypergeom([1 - n, 4/3], [2], 9) fi:
seq(simplify(a(n)), n = 0..21); # Peter Luschny, Mar 30 2023
  • Mathematica
    CoefficientList[Series[1/CubeRoot[(1-9x/(1+x))],{x,0,30}],x] (* Harvey P. Dale, Apr 15 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1+x))^(1/3))

Formula

a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(n-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+2*k) * a(k).
n*a(n) = (7*n-4)*a(n-1) + 8*(n-2)*a(n-2) for n > 1.
a(n) ~ 3^(2/3) * 2^(3*n-1) / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Mar 28 2023
a(n) = (-1)^(1 - n)*3*hypergeom([1 - n, 4/3], [2], 9) for n >= 1. - Peter Luschny, Mar 30 2023

A361882 Expansion of 1/(1 - 9*x/(1 + x)^2)^(1/3).

Original entry on oeis.org

1, 3, 12, 63, 357, 2112, 12834, 79446, 498504, 3160566, 20202882, 129998400, 841084065, 5466859635, 35672889180, 233564188167, 1533744021741, 10097724827904, 66633102118296, 440600483618184, 2918753549183712, 19367330685385032, 128704927930928088
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

Cf. A004987, A180400, A361841, A361842, A361881.
Cf. A361880.

Programs

  • Maple
    a := n -> if n = 0 then 1 else (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) fi: seq(simplify(a(n)), n = 0..22); # Peter Luschny, Mar 30 2023
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1+x)^2)^(1/3))

Formula

a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(n+k-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+2*k) * (n-k) * a(k).
(n-1)*n*a(n) = (7*n-6)*(n-1)*a(n-1) + 6*(n-2)*a(n-2) - (7*n-22)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.
a(n) ~ 3^(1/3) * phi^(4*n) / (Gamma(1/3) * 5^(1/6) * n^(2/3)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 28 2023
a(n) = (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) for n >= 1. - Peter Luschny, Mar 30 2023
Showing 1-3 of 3 results.

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