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.

A371517 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 26, 188, 1459, 11892, 100444, 871528, 7722557, 69590628, 635807180, 5876094308, 54836925779, 516029817620, 4891147100886, 46653935716492, 447490869463145, 4313492172957396, 41763413498670702, 405968522259130636, 3960526930400038404
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(4*k+3, k)/(3*k+4));

Formula

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+3,k)/(3*k+4) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+4,k)/(k+1).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349331.

A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 15, 82, 477, 2901, 18235, 117555, 773085, 5166478, 34987170, 239570655, 1655933060, 11538839130, 80971109712, 571702698185, 4058556404958, 28951715755830, 207424064434502, 1491898838023884, 10768487956456506, 77977009814421534, 566310026687320290
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+2,k)/(2*k+3) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+3,k)/(k+1).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A307678.

A371583 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) / (1 - x) )^2.

Original entry on oeis.org

1, 2, 13, 104, 940, 9166, 94044, 1000602, 10939780, 122161128, 1387361151, 15974899766, 186069556707, 2188416960148, 25953579753464, 310022550197360, 3726709235290628, 45047517497268968, 547217895030263028, 6676784544374859088, 81789906534091716353
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = 2 * Sum_{k=0..n} binomial(5*k+2,k) * binomial(n-1,n-k)/(5*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349332.

A371544 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1+x))^5.

Original entry on oeis.org

1, 5, 30, 220, 1775, 15206, 135745, 1248900, 11758240, 112736305, 1096960024, 10804727805, 107520029780, 1079346767060, 10917110317185, 111149886462926, 1138205538056395, 11715403351807780, 121137702435412040, 1257720947476195045, 13106870738511517659
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,n-k) * binomial(5*k+5,k)/(k+1).
G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349361.
Showing 1-5 of 5 results.