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 1-4 of 4 results.

A369299 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x^3)^3 ).

Original entry on oeis.org

1, 1, 2, 8, 29, 105, 417, 1719, 7181, 30603, 132736, 582790, 2585352, 11575613, 52237278, 237328704, 1084701387, 4983867447, 23007263941, 106658256768, 496336303014, 2317687534865, 10856677523580, 51001805706435, 240225121539000, 1134240896062656, 5367428039668751
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x^3)^3)/x)
    
  • PARI
    a(n, s=3, t=3, u=1) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+k+2,k) * binomial(2*n-3*k,n-3*k).
a(n) = (1/(n+1)) * [x^n] 1/( (1-x) * (1-x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369298 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x^3)^2 ).

Original entry on oeis.org

1, 2, 7, 32, 163, 884, 5011, 29342, 176092, 1077384, 6695093, 42140930, 268108170, 1721372836, 11138994028, 72573587520, 475674650717, 3134297846792, 20750020222815, 137953554890508, 920667400056250, 6165565645765092, 41419898169301995
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x^3)^2)/x)
    
  • PARI
    a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+k+1,k) * binomial(3*n-3*k+1,n-3*k).
a(n) = (1/(n+1)) * [x^n] 1/( (1-x)^2 * (1-x^3)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369399 Expansion of (1/x) * Series_Reversion( x / (1+x) * (1-x^3)^2 ).

Original entry on oeis.org

1, 1, 1, 3, 11, 31, 86, 277, 937, 3095, 10275, 35091, 121662, 423286, 1481648, 5232315, 18601843, 66436069, 238327939, 858805613, 3106856141, 11277393837, 41062303214, 149948280259, 549027748390, 2015108865850, 7412690394406, 27324968423054
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x)*(1-x^3)^2)/x)
    
  • PARI
    a(n, s=3, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial(u*(n+1), n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+k+1,k) * binomial(n+1,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( (1+x) / (1-x^3)^2 )^(n+1). - Seiichi Manyama, Feb 16 2024

A370274 Coefficient of x^n in the expansion of 1/( (1-x) * (1-x^3)^2 )^n.

Original entry on oeis.org

1, 1, 3, 16, 67, 276, 1212, 5391, 24003, 107719, 486728, 2208735, 10059868, 45970367, 210657177, 967636566, 4454109123, 20540731356, 94882599285, 438931979661, 2033217678792, 9429562243530, 43779688919145, 203463271733010, 946445226206940, 4406251540834026
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2024

Keywords

Crossrefs

Cf. A369296.

Programs

  • PARI
    a(n, s=3, t=2, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+k-1,k) * binomial(2*n-3*k-1,n-3*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) * (1-x^3)^2 ). See A369296.
Showing 1-4 of 4 results.