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.

A375316 Expansion of (1 + x)/(1 - x^2*(1 + x)^4).

Original entry on oeis.org

1, 1, 1, 5, 11, 19, 42, 98, 205, 429, 936, 2024, 4316, 9260, 19949, 42841, 91917, 197485, 424331, 911255, 1957086, 4203998, 9029949, 19394681, 41657808, 89478064, 192189304, 412801176, 886657081, 1904452689, 4090567673, 8786123349, 18871714923, 40534539675
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2024

Keywords

Links

Crossrefs

Cf. A093040, A375315.
Cf. A375314.

Programs

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

Formula

a(n) = a(n-2) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(4*k+1,n-2*k).
a(n) = A375314(n) + A375314(n-1).

A375308 a(n) = Sum_{k=0..floor(2*n/3)} binomial(4*n-4*k,2*k).

Original entry on oeis.org

1, 1, 7, 30, 137, 644, 2936, 13625, 62701, 289547, 1335378, 6161329, 28424456, 131135696, 604991601, 2791106585, 12876698159, 59406240678, 274068969337, 1264408966284, 5833313285128, 26911817257385, 124156868897413, 572794023175795, 2642568194952474
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2024

Keywords

Links

Crossrefs

Cf. A108479, A375279, A375307, A375314.

Programs

  • PARI
    a(n) = sum(k=0, 2*n\3, binomial(4*n-4*k, 2*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec((1-x-6*x^2-x^3)/((1-x+2*x^2-x^3)^2-16*x^2))

Formula

a(n) = A375279(2*n).
a(n) = A375314(2*n).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) - 6*a(n-4) + 4*a(n-5) - a(n-6).
G.f.: (1 - x - 6*x^2 - x^3)/((1 - x + 2*x^2 - x^3)^2 - 16*x^2).

A375318 Expansion of 1/(1 - x^3*(1 + x)^4).

Original entry on oeis.org

1, 0, 0, 1, 4, 6, 5, 9, 28, 57, 82, 122, 249, 519, 913, 1485, 2632, 5053, 9369, 16375, 28662, 52226, 96182, 173220, 307653, 551927, 1002327, 1815191, 3258813, 5845015, 10539893, 19048900, 34332648, 61735922, 111129005, 200406479, 361364501, 650804074, 1171717523
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2024

Keywords

Links

Crossrefs

Cf. A115055, A264622.
Cf. A375314.

Programs

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

Formula

a(n) = a(n-3) + 4*a(n-4) + 6*a(n-5) + 4*a(n-6) + a(n-7).
a(n) = Sum_{k=0..floor(n/3)} binomial(4*k,n-3*k).
Showing 1-3 of 3 results.

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