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 1-4 of 4 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

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).

A378690 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(7/2)/(1 - x*A(x)) )^2.

Original entry on oeis.org

1, 2, 17, 190, 2438, 33938, 498413, 7602010, 119261202, 1912171310, 31194947785, 516153663072, 8641160417191, 146105874059670, 2491396820758004, 42795782630083868, 739842609794223330, 12862556429464405500, 224744883747568868574, 3944534317072930309360
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=2, s=1, t=7, u=2) = 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)^3/(1 - x*A(x)) )^2.
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A378688.
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).

A378685 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)^3).

Original entry on oeis.org

1, 1, 8, 88, 1126, 15716, 232069, 3564835, 56382489, 912031280, 15018257510, 250913307393, 4242722219425, 72470224174650, 1248608968982903, 21673752440979879, 378677335852165297, 6654158090059397480, 117523324766568499072, 2085095374834405245007
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=1, 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)^6/(1 - x*A(x)^3)).
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).

A378693 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(7/6)/(1 - x*A(x)) )^6.

Original entry on oeis.org

1, 6, 63, 806, 11445, 173388, 2745470, 44891118, 752141682, 12845874594, 222813745704, 3914269052736, 69501455945987, 1245309605501088, 22488056019050124, 408861223600687710, 7478056231521533658, 137496627558561863460, 2540015518588821201453
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=6, 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)/(1 - x*A(x)) )^6.
G.f.: A(x) = B(x)^6 where B(x) is the g.f. of A378694.
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 1-4 of 4 results.