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 1-9 of 9 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).

A180399 Expansion of (1/3)*(1 - (1-9*x-9*x^2)^(1/3)).

Original entry on oeis.org

0, 1, 4, 21, 138, 999, 7683, 61542, 507663, 4281849, 36748998, 319845591, 2816007714, 25032803841, 224355173193, 2024955168606, 18388543939947, 167882583075453, 1540000362501702, 14186252492098011, 131176523761136568, 1217094112710349731, 11327464549934673309
Offset: 0

Views

Author

Clark Kimberling, Sep 01 2010

Keywords

Examples

			The Maclaurin series begins with x + 4x^2 + 21x^3.

Links

Crossrefs

Cf. A180400.

Programs

  • Mathematica
    CoefficientList[Series[1/3*(1-(1-9*x-9*x^2)^(1/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
  • PARI
    x='x+O('x^66); concat([0],Vec(1/3*(1-(1-9*x-9*x^2)^(1/3)))) \\ Joerg Arndt, Jun 01 2013

Formula

G.f.: (1/3)*(1 - (1-9*x-9*x^2)^(1/3)).
a(n) = sum(m=1..n, binomial(m,n-m)/m * sum(k=0..m-1, binomial(k,m-1-k) * 3^k*(-1)^(m-1-k) * binomial(m+k-1,m-1))). [From Vladimir Kruchinin, Feb 08 2011]
Recurrence: n*a(n) = 3*(3*n-4)*a(n-1) + 3*(3*n-8)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ ((13-3*sqrt(13))/2)^(1/3)/(9*Gamma(2/3)) * ((9+3*sqrt(13))/2)^n/(n^(4/3)). - Vaclav Kotesovec, Oct 20 2012

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

A376568 Expansion of 1/(1 - 9*x*(1 + x))^(2/3).

Original entry on oeis.org

1, 6, 51, 450, 4095, 37908, 354978, 3351348, 31833945, 303822090, 2910657321, 27970777926, 269484894081, 2602002636540, 25170322256010, 243876058527132, 2366251795228437, 22987502934573762, 223563791480714685, 2176402892261301990, 21206170582394740371
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A180400, A377260, A377261.
Cf. A377233.

Programs

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

Formula

a(n) = 3*((3*n-1)*a(n-1) + (3*n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-2/3,k) * binomial(k,n-k).
a(n) ~ (3 + sqrt(13))^(n + 2/3) * 3^n / (Gamma(2/3) * 13^(1/3) * n^(1/3) * 2^(n + 2/3)). - Vaclav Kotesovec, Oct 26 2024

A377260 Expansion of 1/(1 - 9*x*(1 + x))^(4/3).

Original entry on oeis.org

1, 12, 138, 1512, 16191, 170856, 1785042, 18514548, 190978047, 1961435736, 20074741596, 204870399552, 2085761241018, 21191569851312, 214930928188116, 2176565295933000, 22012171108148025, 222351327936731700, 2243667436429422150, 22618648367553735000, 227826739721910301245
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A180400, A376568, A377261.
Cf. A377234.

Programs

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

Formula

a(n) = 3*((3*n+1)*a(n-1) + (3*n+2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-4/3,k) * binomial(k,n-k).
a(n) ~ n^(1/3) * 3^(n+1) * (3 + sqrt(13))^(n + 4/3) / (13^(2/3) * Gamma(1/3) * 2^(n + 4/3)). - Vaclav Kotesovec, May 03 2025

A377261 Expansion of 1/(1 - 9*x*(1 + x))^(5/3).

Original entry on oeis.org

1, 15, 195, 2340, 26910, 301158, 3307590, 35830080, 384072975, 4082949585, 43113860361, 452742067440, 4732188244290, 49266375442110, 511157395433610, 5287689996408612, 54555878321808435, 561579617798527185, 5768783256563735265, 59149668761521664040, 605472238745163334116
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A180400, A376568, A377260.
Cf. A377235.

Programs

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

Formula

a(n) = 3*((3*n+2)*a(n-1) + (3*n+4)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-5/3,k) * binomial(k,n-k).
a(n) ~ Gamma(1/3) * n^(2/3) * 3^(n + 3/2) * (3 + sqrt(13))^(n + 5/3) / (Pi * 13^(5/6) * 2^(n + 11/3)). - Vaclav Kotesovec, May 03 2025

A372087 G.f. A(x) satisfies A(x) = 1/( 1 - 9*x*(1 + x)*A(x) )^(1/3).

Original entry on oeis.org

1, 3, 30, 369, 5130, 76626, 1200816, 19475829, 324140886, 5504511654, 94998663000, 1661370690546, 29377608173460, 524366947411668, 9435112261205328, 170958245619049173, 3116653690408787070, 57125853834377116014, 1052116816793294021688
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2024

Keywords

Crossrefs

Cf. A180400, A372038.

Programs

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

Formula

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

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 1-9 of 9 results.

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