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 1-10 of 10 results.

A383597 Expansion of 1/( (1-x)^2 * (1-10*x) )^(1/3).

Original entry on oeis.org

1, 4, 25, 190, 1570, 13552, 120178, 1085620, 9940345, 91962460, 857750233, 8053389142, 76026759760, 721017894640, 6864725124520, 65578937628304, 628320730656586, 6035594205744520, 58110220504754650, 560624083417180300, 5418599393597801020, 52459116546784350880
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A383598, A383599.
Cf. A004987, A361375, A376802, A383601.

Programs

  • Magma
    I:=[4,25]; [1] cat [n le 2 select I[n] else ((11*n-7)*Self(n-1) - 10*(n-1) *Self(n-2))/n : n in [1..30]]; // Vincenzo Librandi, May 04 2025
  • Mathematica
    Table[Sum[(-9)^k *Binomial[-1/3,k]* Binomial[n, k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, May 04 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-1/3, k)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n,k).
n*a(n) = (11*n-7)*a(n-1) - 10*(n-1)*a(n-2) for n > 1.
a(n) ~ 10^(n + 2/3) / (Gamma(1/3) * 3^(4/3) * n^(2/3)). - Vaclav Kotesovec, May 02 2025
a(n) = hypergeom([1/3, -n], [1], -9). - Stefano Spezia, May 04 2025

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).

A377233 Expansion of 1/(1 - 9*x/(1-x))^(2/3).

Original entry on oeis.org

1, 6, 51, 456, 4191, 39174, 370329, 3529284, 33838854, 325978044, 3152058630, 30572797920, 297294956070, 2897207397420, 28286321963370, 276611636831640, 2708781551458665, 26559205696513590, 260695647288540915, 2561413004129212440, 25188928968792165495
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A052268, A361375, A377234, A377235.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-2/3, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..n-1} (2+k/n) * a(k).
a(n) = ((11*n-5)*a(n-1) - 10*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-2/3,k) * binomial(n-1,n-k).
a(n) ~ Gamma(1/3) * 3^(11/6) * 2^(n - 5/3) * 5^(n - 2/3) / (Pi * n^(1/3)). - Vaclav Kotesovec, Oct 21 2024
a(n) = 6*hypergeom([5/3, 1-n], [2], -9) for n > 0. - Stefano Spezia, May 04 2025

A377234 Expansion of 1/(1 - 9*x/(1-x))^(4/3).

Original entry on oeis.org

1, 12, 138, 1524, 16455, 175152, 1846164, 19320456, 201093843, 2084105820, 21524823858, 221678089716, 2277558628869, 23352604052952, 239024756624520, 2442818071519104, 24932208295715538, 254166614639215032, 2588333499216072516, 26333774228774140680, 267693203735009601870
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A052268, A361375, A377233, A377235.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-4/3, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..n-1} (4-k/n) * a(k).
a(n) = ((11*n+1)*a(n-1) - 10*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-4/3,k) * binomial(n-1,n-k).
a(n) ~ 3^(11/3) * 10^(n - 4/3) * n^(1/3) / Gamma(1/3). - Vaclav Kotesovec, Oct 21 2024
a(n) = 12*hypergeom([7/3, 1-n], [2], -9) for n > 0. - Stefano Spezia, May 04 2025

A377235 Expansion of 1/(1 - 9*x/(1-x))^(5/3).

Original entry on oeis.org

1, 15, 195, 2355, 27285, 307833, 3409485, 37253805, 402847620, 4320615390, 46032234486, 487743084150, 5144152999650, 54041442437850, 565803538944450, 5906360704312770, 61495776957754725, 638808193722602175, 6622218378818049075, 68522901145021162275, 707856527414874575805
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A052268, A361375, A377233, A377234.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-5/3, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..n-1} (5-2*k/n) * a(k).
a(n) = ((11*n+4)*a(n-1) - 10*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-5/3,k) * binomial(n-1,n-k).
a(n) ~ Gamma(1/3) * 3^(29/6) * 2^(n - 11/3) * 5^(n - 5/3) * n^(2/3) / Pi. - Vaclav Kotesovec, Oct 21 2024
a(n) = 15*hypergeom([8/3, 1-n], [2], -9) for n > 0. - Stefano Spezia, May 04 2025

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

A372110 G.f. A(x) satisfies A(x) = ( (1 - x*A(x))/(1 - 10*x*A(x)) )^(1/3).

Original entry on oeis.org

1, 3, 30, 381, 5457, 84000, 1356726, 22680705, 389100000, 6811276449, 121177168266, 2184600000000, 39822674320065, 732762138176436, 13592289000000000, 253896500477864361, 4771765283550516435, 90167361600000000000, 1712019315455953465026
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2024

Keywords

Links

Crossrefs

Cf. A001003, A372109.
Cf. A361375, A372019, A372091, A372105, A378552.

Programs

  • PARI
    a(n) = sum(k=0, n, 9^k*binomial(n/3+k-2/3, k)*binomial(n-1, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 9^k * binomial(n/3+k-2/3,k) * binomial(n-1,n-k).
From Seiichi Manyama, Nov 30 2024: (Start)
G.f.: exp( Sum_{k>=1} A378552(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - 9*x/(1-x))^((n+1)/3).
G.f.: (1/x) * Series_Reversion( x*(1 - 9*x/(1-x))^(1/3) ). (End)

A372105 G.f. A(x) satisfies A(x) = 1/( 1 - 9*x*A(x)/(1-x) )^(1/3).

Original entry on oeis.org

1, 3, 30, 372, 5187, 77682, 1220421, 19842492, 331047138, 5635390827, 97491744102, 1709067567387, 30293505154362, 542010120629052, 9775907758416999, 177556754470826046, 3244682279388126576, 59614554914471797482, 1100573677912355195313
Offset: 0

Views

Author

Seiichi Manyama, Apr 18 2024

Keywords

Crossrefs

Cf. A361375, A372087.

Programs

  • PARI
    a(n) = sum(k=0, n, 9^k*binomial(4*k/3-2/3, k)*binomial(n-1, n-k)/(k+1));

Formula

a(n) = Sum_{k=0..n} 9^k * binomial(4*k/3-2/3,k) * binomial(n-1,n-k)/(k+1).
Showing 1-10 of 10 results.

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