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 1-5 of 5 results.

A376729 Expansion of (1 - x^2 - x^3)/((1 - x^2 - x^3)^2 - 4*x^5).

Original entry on oeis.org

1, 0, 1, 1, 1, 6, 2, 15, 16, 29, 71, 73, 212, 276, 541, 1016, 1497, 3189, 4825, 9162, 16022, 26763, 50424, 82869, 151851, 262705, 456520, 820328, 1401913, 2511824, 4361521, 7657481, 13528913, 23509678, 41633002, 72630919, 127709888, 224418509, 392539055, 691382201
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2024

Keywords

Links

Crossrefs

Cf. A108479, A376730, A376731.
Cf. A376723, A376726.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-x^2-x^3)/((1-x^2-x^3)^2-4*x^5))
  • PARI
    a(n) = sum(k=0, n\2, binomial(2*k, 2*n-4*k));

Formula

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k,2*n-4*k).

A376723 Expansion of 1/((1 - x^2 - x^3)^2 - 4*x^5).

Original entry on oeis.org

1, 0, 2, 2, 3, 10, 7, 28, 33, 64, 132, 170, 408, 578, 1119, 2002, 3194, 6310, 10021, 18666, 32353, 55450, 101443, 170672, 308744, 534820, 935936, 1663892, 2872669, 5111652, 8898082, 15641802, 27538647, 48049562, 84813451, 148219128, 260572901, 457451088
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Links

Crossrefs

Cf. A182890, A376724, A376725.
Cf. A376726, A376729.
Cf. A375278.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/((1-x^2-x^3)^2-4*x^5))
  • PARI
    a(n) = sum(k=0, n\2, binomial(2*k+2, 2*n-4*k+1))/2;

Formula

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).
a(n) = (1/2) * Sum_{k=0..floor(n/2)} binomial(2*k+2,2*n-4*k+1).

A376727 Expansion of (1 + x^3 - x^4)/((1 + x^3 - x^4)^2 - 4*x^3).

Original entry on oeis.org

1, 0, 0, 3, 1, 0, 5, 10, 1, 7, 35, 21, 10, 84, 126, 47, 166, 462, 343, 341, 1288, 1731, 1170, 3081, 6453, 5685, 7553, 19572, 25280, 24004, 52789, 93844, 95932, 143435, 299577, 386536, 448673, 873754, 1411193, 1625003, 2536215, 4639077, 6097214, 7959492, 14238226
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2024

Keywords

Links

Crossrefs

Cf. A099511, A376726, A376728.
Cf. A376724, A376730.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec((1+x^3-x^4)/((1+x^3-x^4)^2-4*x^3))
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*k+1, 2*n-6*k+1));

Formula

a(n) = 2*a(n-3) + 2*a(n-4) - a(n-6) + 2*a(n-7) - a(n-8).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k+1,2*n-6*k+1).

A376728 Expansion of (1 + x^4 - x^5)/((1 + x^4 - x^5)^2 - 4*x^4).

Original entry on oeis.org

1, 0, 0, 0, 3, 1, 0, 0, 5, 10, 1, 0, 7, 35, 21, 1, 9, 84, 126, 36, 12, 165, 462, 330, 68, 287, 1287, 1716, 730, 533, 3004, 6435, 5022, 2045, 6293, 19449, 24329, 13345, 14008, 50524, 92400, 76912, 47481, 120156, 294124, 354488, 237139, 299421, 823200, 1354588
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2024

Keywords

Links

Crossrefs

Cf. A099511, A376726, A376727.
Cf. A376725, A376731.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec((1+x^4-x^5)/((1+x^4-x^5)^2-4*x^4))
  • PARI
    a(n) = sum(k=0, n\4, binomial(2*k+1, 2*n-8*k+1));

Formula

a(n) = 2*a(n-4) + 2*a(n-5) - a(n-8) + 2*a(n-9) - a(n-10).
a(n) = Sum_{k=0..floor(n/4)} binomial(2*k+1,2*n-8*k+1).

A376787 Expansion of (1 - x^2 + x^3)/((1 - x^2 + x^3)^2 - 4*x^3).

Original entry on oeis.org

1, 0, 1, 3, 1, 10, 6, 21, 36, 43, 127, 139, 340, 540, 881, 1832, 2653, 5427, 8829, 15550, 28642, 46805, 87756, 147575, 262751, 465591, 797864, 1437816, 2471553, 4383696, 7689305, 13402819, 23752217, 41305842, 72916606, 127708213, 223809012, 394045411
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2024

Keywords

Links

Crossrefs

Cf. A376716, A376788.
Cf. A376723, A376726, A376729.

Programs

  • Mathematica
    CoefficientList[Series[(1-x^2+x^3)/((1-x^2+x^3)^2-4x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,2,-1,2,-1},{1,0,1,3,1,10},40] (* Harvey P. Dale, Aug 11 2025 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-x^2+x^3)/((1-x^2+x^3)^2-4*x^3))
  • PARI
    a(n) = sum(k=0, n\2, binomial(2*k+1, 2*n-4*k));

Formula

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

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