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.

A368968 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 4, 26, 206, 1813, 17030, 167229, 1695920, 17624932, 186722580, 2009077416, 21894695420, 241170873096, 2680761546396, 30032284769832, 338744791093796, 3843699928567438, 43844993166845920, 502497843180361288, 5783367971991398760, 66815895492710846218
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x-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((t+u+1)*(n+1)-(s-1)*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(5*n-2*k+3,n-3*k).

A368957 Expansion of (1/x) * Series_Reversion( x * (1-x^2/(1-x))^2 ).

Original entry on oeis.org

1, 0, 2, 2, 13, 28, 127, 376, 1522, 5210, 20403, 74952, 292313, 1114704, 4371839, 17040586, 67378981, 266402370, 1061919289, 4241539218, 17030430061, 68554148388, 276988107861, 1121954081852, 4557637048543, 18556386241468, 75729621399950
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(n-k-1,n-2*k).
D-finite with recurrence 2000*n*(48911424697856946605*n -85862091501967897127)*(2*n+1) *(2*n-1)*(n+1)*a(n) +20*n*(2*n-1) *(9782284939571389321000*n^3 -124853950521493511435497*n^2 +291346534864358121613940*n -174094174192357320452243)*a(n-1) +6*(-1056620466555214160730036*n^5 +5240184994626612582867927*n^4 -10842595636486250859803566*n^3 +12555800263623324081669713*n^2 -8323849827256795107408998*n +2408908212964334471344960)*a(n-2) +(-11765946248792268093670721*n^5 +111908835475719217483707009*n^4 -409273054609037480568616913*n^3 +706828511197147489881004671*n^2 -556026097737885029117618846*n +145005575225258917734060720)*a(n-3) +12*(110108843793156901781209*n^5 -1706708924562157727758594*n^4 +10728825545391547292463142*n^3 -34121900584137543620498771*n^2+54762746448568812780284884*n -35381689886652975706836240)*a(n-4) -36*(3*n-11)*(n-4)*(3*n-13) *(2*n-7)*(36626509829570139536*n -97211536327074911575)*a(n-5)=0. - R. J. Mathar, Jan 25 2024

A369014 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^3 ).

Original entry on oeis.org

1, 0, 0, 3, 3, 3, 36, 78, 129, 685, 2043, 4554, 17233, 57279, 153045, 509848, 1724739, 5117643, 16445555, 55165536, 173225715, 555899673, 1847495415, 5971507824, 19333284247, 63975307425, 209807070669, 685973054145, 2269660792842, 7501194321663, 24725092907853
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3/(1-x))^3)/x)
    
  • PARI
    a(n, s=3, t=3, u=-3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*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(n-2*k-1,n-3*k).

A369077 Expansion of (1/x) * Series_Reversion( x * (1+x^3/(1-x))^2 ).

Original entry on oeis.org

1, 0, 0, -2, -2, -2, 13, 32, 55, -72, -439, -1152, -506, 4870, 20613, 31744, -26392, -313096, -826529, -654362, 3635175, 16431826, 30100349, -15474300, -262654439, -780688624, -756130333, 3013376172, 15711713509, 31584466782, -6090973971, -250819494954
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x^3/(1-x))^2)/x)
    
  • PARI
    a(n, s=3, t=2, u=-2) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

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

A369489 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 26, 98, 387, 1589, 6688, 28676, 124880, 550926, 2456831, 11056693, 50152457, 229050621, 1052393802, 4861062466, 22559964766, 105144660498, 491922058878, 2309456782464, 10876596029574, 51372213424194, 243283513468707, 1154929327702775, 5495105429597720
Offset: 0

Views

Author

Seiichi Manyama, Jan 24 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1-x)*(1-x-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((t-u+1)*(n+1)-(s-1)*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(2*n-2*k,n-3*k).
Showing 1-5 of 5 results.