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.

A371613 G.f. satisfies A(x) = ( 1 + x / (1 - x*A(x)^3) )^2.

Original entry on oeis.org

1, 2, 3, 16, 83, 460, 2767, 17210, 110308, 723624, 4832363, 32747106, 224619408, 1556484636, 10879744696, 76621739626, 543159825499, 3872610857558, 27752175177823, 199787917082084, 1444171829169939, 10477887409768628, 76275565075016394
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=2, s=1, t=0, u=6) = 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) = Sum_{k=0..n} binomial(6*(n-k)+2,k) * binomial(n-1,n-k)/(3*(n-k)+1).

A371607 G.f. satisfies A(x) = ( 1 + x * (1 + x*A(x)^2) )^2.

Original entry on oeis.org

1, 2, 3, 10, 29, 92, 314, 1078, 3830, 13844, 50746, 188554, 707667, 2679960, 10227940, 39294772, 151859858, 589943516, 2302462140, 9023681820, 35498194465, 140122652960, 554827907272, 2203135245820, 8771143399104, 35003747271444, 140002994665366
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=2, s=1, t=0, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

a(n) = Sum_{k=0..n} binomial(4*(n-k)+2,k) * binomial(k,n-k)/(2*(n-k)+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A137954.

A371614 G.f. satisfies A(x) = ( 1 + x / (1 - x*A(x)^2)^2 )^2.

Original entry on oeis.org

1, 2, 5, 26, 138, 814, 5051, 32550, 215792, 1461934, 10077345, 70450980, 498328320, 3559894566, 25646621725, 186122575840, 1359384244220, 9984580141702, 73703387448245, 546492958156148, 4068417329371228, 30397841636794944, 227872480308702892
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=2, s=2, t=0, u=4) = 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) = Sum_{k=0..n} binomial(4*(n-k)+2,k) * binomial(n+k-1,n-k)/(2*(n-k)+1).

A378731 G.f. A(x) satisfies A(x) = ( 1 + x / (1 - x*A(x)^(4/3)) )^3.

Original entry on oeis.org

1, 3, 6, 22, 93, 417, 1993, 9864, 50217, 261239, 1382448, 7418877, 40278175, 220830513, 1220930337, 6799458685, 38107621704, 214771481163, 1216457185122, 6920603372448, 39529745832681, 226605757331904, 1303291125124071, 7518151040142000, 43488151271999326
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=3, s=1, t=0, u=4) = 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

G.f.: A(x) = (1 + x*B(x))^3 where B(x) is the g.f. of A364743.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = 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).

A378732 G.f. A(x) satisfies A(x) = ( 1 + x / (1 - x*A(x)) )^4.

Original entry on oeis.org

1, 4, 10, 36, 155, 704, 3384, 16844, 86097, 449344, 2384170, 12822556, 69743953, 382982940, 2120323014, 11822279232, 66327376437, 374162700460, 2120999728610, 12075668658000, 69021358842795, 395909382981572, 2278286453089574, 13149207655326372, 76096242994616990
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=4, s=1, t=0, u=4) = 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

G.f.: A(x) = (1 + x*B(x))^4 where B(x) is the g.f. of A364743.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = 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).
Showing 1-5 of 5 results.