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 1-3 of 3 results.

A378692 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)).

Original entry on oeis.org

1, 1, 8, 86, 1075, 14667, 211799, 3182454, 49243854, 779379652, 12558073022, 205312307834, 3397359326116, 56790504859929, 957574385205771, 16267419813629731, 278162968238908681, 4783813617177604232, 82691541747420586716, 1435895455224032519430, 25035634270828781060188
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A000108, A001003, A108447, A364747, A364748, A378691.
Cf. A378686, A378690, A378693.
Cf. A378685, A378688, A378694.

Programs

  • PARI
    a(n, r=1, s=1, t=7, u=1) = 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) satisfies A(x) = 1/(1 - x*A(x)^6/(1 - x*A(x))).
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).

A378686 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(7/3)/(1 - x*A(x)) )^3.

Original entry on oeis.org

1, 3, 27, 313, 4122, 58584, 875897, 13577139, 216224616, 3516601243, 58160887857, 975211608399, 16539799297342, 283243124783136, 4890858070498203, 85060240453556192, 1488653675438168001, 26197808077514204832, 463311206395709908936, 8229849868810254813378
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A108447, A371581.
Cf. A349017, A369012.
Cf. A378690, A378692, A378693.
Cf. A378685.

Programs

  • PARI
    a(n, r=3, s=1, t=7, u=3) = 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) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)) )^3.
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A378685.
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).

A378694 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)^6).

Original entry on oeis.org

1, 1, 8, 91, 1210, 17577, 270314, 4326070, 71300386, 1202012254, 20630488004, 359279348424, 6332747550808, 112761701957119, 2025325557546780, 36650364776763804, 667570101840389826, 12229542931765845994, 225183117821591853440, 4165207037639796288385
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A378685, A378688, A378692.

Programs

  • PARI
    a(n, r=1, s=1, t=7, 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

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^6/(1 - x*A(x)^6)).
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 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.