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.

A371521 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^6.

Original entry on oeis.org

1, 6, 57, 614, 7158, 88002, 1123689, 14760024, 198172050, 2707560544, 37522666803, 526190125308, 7452866846847, 106465245105972, 1532129408941797, 22191180837313808, 323243244688652943, 4732225866305323686, 69591395772704207770, 1027547992261749954798
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 6*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+5, k)/(5*k+6));

Formula

a(n) = 6 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+5,k)/(5*k+6) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+6,k)/(k+1).
G.f.: A(x) = B(x)^6 where B(x) is the g.f. of A349333.

A371523 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3 / (1-x))^2.

Original entry on oeis.org

1, 2, 15, 142, 1533, 17924, 220936, 2827218, 37202580, 500228562, 6842899886, 94931338876, 1332438761910, 18887047322030, 269986427261981, 3887654399820062, 56337997080499605, 821021578186212094, 12024687038651388155, 176900548019426869808, 2612917215947948178941
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 2*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+1, k)/(5*k+2));

Formula

a(n) = 2 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+1,k)/(5*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349333.

A371379 G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/2) / (1-x))^4.

Original entry on oeis.org

1, 4, 34, 344, 3859, 46240, 579722, 7511272, 99782617, 1351784792, 18604380884, 259395030992, 3656180724752, 52011780756632, 745799171500502, 10768038899631476, 156414710103922340, 2284233700081510820, 33517461646190624690, 493917761019513208800
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+3, k)/(5*k+4));

Formula

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+3,k)/(5*k+4).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349333.

A371519 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 45, 470, 5375, 65231, 825225, 10764185, 143739440, 1955340360, 27001732972, 377530388235, 5333865386885, 76031188364860, 1092117166466660, 15792298241897649, 229704197116753825, 3358528175751886765, 49333470827844265285, 727680248026484478405
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+4, k)/(k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+4,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^5 ).
G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349333.

A371538 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1+x))^3.

Original entry on oeis.org

1, 3, 18, 151, 1440, 14835, 160793, 1806849, 20859129, 245905348, 2947869600, 35825319390, 440372147956, 5465555197818, 68396554601013, 862066323857486, 10933638171672105, 139439595024315675, 1787056241039876890, 23003636498360053905, 297283046361025602900
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = 3 * Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,n-k) * binomial(6*k+3,k)/(6*k+3).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349362.
Showing 1-5 of 5 results.