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.

A366034 G.f. A(x) satisfies A(x) = 1/(1 - x)^3 + x*(1 - x)^3*A(x)^3.

Original entry on oeis.org

1, 4, 15, 70, 405, 2676, 19075, 142562, 1100961, 8711968, 70257055, 575269278, 4769615773, 39961571228, 337805166747, 2877506096794, 24675158973081, 212835736433304, 1845348003175063, 16073746202176150, 140590118902532757, 1234285061013293716
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2023

Keywords

Crossrefs

Programs

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

Formula

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

A366182 G.f. A(x) satisfies A(x) = 1/(1 - x)^3 + x*A(x)^3/(1 - x).

Original entry on oeis.org

1, 4, 19, 128, 1037, 9221, 86847, 851073, 8586951, 88598014, 930473246, 9913648325, 106891041270, 1164153791878, 12788021717902, 141518588447588, 1576271179332762, 17657110535606919, 198792746866201879, 2248222906227731856, 25529220583699163958
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2023

Keywords

Crossrefs

Partial sums of A366180.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+6*k+2, n-k)*binomial(3*k, k)/(2*k+1));

Formula

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

A366183 G.f. A(x) satisfies A(x) = 1/(1 - x)^3 + x*A(x)^3/(1 - x)^2.

Original entry on oeis.org

1, 4, 20, 145, 1250, 11746, 116641, 1204039, 12790067, 138895021, 1535005454, 17207743738, 195197256289, 2236419124408, 25842382083071, 300822398531482, 3524358836945936, 41524956284752018, 491722951928324392, 5848997420625891294, 69854562522309219081
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+7*k+2, n-k)*binomial(3*k, k)/(2*k+1));

Formula

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

A376159 G.f. satisfies A(x) = 1 / ((1-x)^3 - x*A(x)).

Original entry on oeis.org

1, 4, 17, 90, 539, 3451, 23100, 159720, 1131905, 8178326, 60019533, 446166771, 3352530190, 25422458170, 194302002463, 1495223230621, 11575504625874, 90090318248607, 704480581789900, 5532228951823605, 43610427926723780, 344972119634359080, 2737451123900901555
Offset: 0

Views

Author

Seiichi Manyama, Sep 12 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^3+sqrt((1-x)^6-4*x)))
    
  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k+2, n-k)*binomial(2*k, k)/(k+1));

Formula

G.f.: 2 / ((1-x)^3 + sqrt((1-x)^6 - 4*x)).
a(n) = Sum_{k=0..n} binomial(n+5*k+2,n-k) * binomial(2*k,k)/(k+1).

A376160 G.f. satisfies A(x) = 1 / ((1-x)^3 - x*A(x)^3).

Original entry on oeis.org

1, 4, 25, 260, 3205, 42966, 609567, 8999164, 136811781, 2127343669, 33675622992, 540878965522, 8792433396559, 144383416380703, 2391557494237062, 39910530610590312, 670383542665237001, 11325278943044058378, 192301381444863249559, 3280101940070399446926
Offset: 0

Views

Author

Seiichi Manyama, Sep 12 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+11*k+2, n-k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+11*k+2,n-k) * binomial(4*k,k)/(3*k+1).
Showing 1-5 of 5 results.