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 1-6 of 6 results.

A361375 Expansion of 1/(1 - 9*x/(1 - x))^(1/3).

Original entry on oeis.org

1, 3, 21, 165, 1380, 11982, 106626, 965442, 8854725, 82022115, 765787773, 7195638909, 67973370618, 644991134880, 6143707229880, 58714212503784, 562741793028282, 5407273475087934, 52074626299010130, 502513862912425650, 4857975310180620720
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

Cf. A004987, A361843, A361844, A361845, A361880, A361895, A361896.
Cf. A085362.

Programs

  • Maple
    a := n -> if n = 0 then 1 else 3*hypergeom([1 - n, 4/3], [2], -9) fi:
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 30 2023
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x))^(1/3))

Formula

a(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} (n+2*k) * a(k).
n*a(n) = (11*n-8)*a(n-1) - 10*(n-2)*a(n-2) for n > 1.
a(n) ~ 3^(2/3) * 10^(n - 1/3) / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Mar 28 2023
a(n) = 3*hypergeom([1 - n, 4/3], [2], -9) for n >= 1. - Peter Luschny, Mar 30 2023

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).

A361880 Expansion of 1/(1 - 9*x/(1 - x)^2)^(1/3).

Original entry on oeis.org

1, 3, 24, 207, 1893, 17952, 174402, 1723494, 17250000, 174354822, 1776119970, 18208500000, 187659221409, 1942674634371, 20187543581880, 210472842939975, 2200677521078253, 23068297001178240, 242353695578011416, 2551260130246575048, 26905595698893121728
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

Cf. A004987, A361375, A361843, A361844, A361845, A361895, A361896.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x)^2)^(1/3))

Formula

a(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} (n+2*k) * (n-k) * a(k).
(n-1)*n*a(n) = (11*n-6)*(n-1)*a(n-1) - 18*(n-2)*a(n-2) - (11*n-38)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.
a(n) ~ 3^(1/3) * ((11 + 3*sqrt(13))/2)^n / (Gamma(1/3) * 13^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 28 2023

A361895 Expansion of 1/(1 - 9*x/(1 - x)^3)^(1/3).

Original entry on oeis.org

1, 3, 27, 252, 2487, 25434, 266364, 2837082, 30601233, 333302931, 3658565127, 40413860334, 448778693844, 5005642415907, 56044616215041, 629552293867800, 7092072533703567, 80095810435943526, 906605837653876254, 10282430320166723448, 116829834042508121682
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

Cf. A004987, A361375, A361843, A361844, A361845, A361880, A361896.

Programs

  • Mathematica
    a[0]=1; a[n_]:=3*n*(1 + n)*HypergeometricPFQ[{1-n, 1+n/2, (3+n)/2}, {5/3, 2}, -4/3]/2; Array[a,21,0] (* Stefano Spezia, May 02 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x)^3)^(1/3))

Formula

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+2*k-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * binomial(n+1-k,2) * a(k).
a(n) = 3*n*(1 + n)*hypergeom([1-n, 1+n/2, (3+n)/2], [5/3, 2], -4/3)/2 for n > 0. - Stefano Spezia, May 02 2024
a(n) ~ ((7 - sqrt(21))^(1/3) + (7 + sqrt(21))^(1/3))^(1/3) * (4 + (3*((39 - sqrt(21))/2))^(1/3) + (3*((39 + sqrt(21))/2))^(1/3))^n / (Gamma(1/3) * 2^(1/9) * 7^(2/9) * n^(2/3)). - Vaclav Kotesovec, Jul 11 2025

A361896 Expansion of 1/(1 - 9*x/(1 - x)^4)^(1/3).

Original entry on oeis.org

1, 3, 30, 300, 3165, 34584, 386880, 4400928, 50692266, 589584042, 6910397886, 81507086634, 966408021984, 11509174498254, 137584249375308, 1650109151463594, 19847075122106145, 239316542492974317, 2892135259684291248, 35021199836282568456, 424837125616822551264
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

Cf. A004987, A361375, A361843, A361844, A361845, A361880, A361895.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x)^4)^(1/3))

Formula

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+3*k-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * binomial(n+2-k,3) * a(k).

A372039 Expansion of ( 1 + 9*x*(1 + x) )^(1/3).

Original entry on oeis.org

1, 3, -6, 27, -144, 837, -5139, 32778, -215001, 1440747, -9818820, 67834665, -473945580, 3342743235, -23766448545, 170148578130, -1225477405485, 8873126329095, -64547392633740, 471509782020405, -3457212506428230, 25434642838306185, -187694935991201745
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2024

Keywords

Crossrefs

Cf. A180400, A361843, A372024, A372038.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec((1+9*x*(1+x))^(1/3))
  • PARI
    a(n) = sum(k=0, n, 9^k*binomial(1/3, k)*binomial(k, n-k));

Formula

a(n) = Sum_{k=0..n} 9^k * binomial(1/3,k) * binomial(k,n-k).
a(n) ~ (-1)^(n+1) * Gamma(1/3) * 5^(1/6) * 3^(n - 1/2) * phi^(2*n - 2/3) / (2*Pi*n^(4/3)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 19 2024
Showing 1-6 of 6 results.

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