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.

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

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).

A349018 G.f. A(x) satisfies A(x) = 1/(1 - x/(1 - x*A(x)))^4.

Original entry on oeis.org

1, 4, 14, 60, 297, 1584, 8868, 51412, 305964, 1858308, 11472152, 71774548, 454080514, 2899959640, 18670920458, 121056521536, 789733186076, 5180002637472, 34141018474400, 225995779077324, 1501809350268648, 10015202238242356, 67003372168525774
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2021

Keywords

Crossrefs

Programs

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

Formula

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
a(n) ~ sqrt((1 - r*s)*(1 - r - r*s) / (2 - r*(2*s - 3))) / (sqrt(2*Pi) * n^(3/2) * r^(n+1)), where r = 0.13968480593491705709394976139265608086009606657813769... and s = 3.10146641162846907900664383717504887133026560522911567... are real roots of the system of equations (-1 + r*s)^4/(-1 + r + r*s)^4 = s, (4*r^2*(-1 + r*s)^3)/(-1 + r + r*s)^5 = 1. - Vaclav Kotesovec, Nov 15 2021
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A364723. - Seiichi Manyama, Dec 04 2024

A378882 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)/(1 - x*A(x)^(5/3)) )^3.

Original entry on oeis.org

1, 3, 15, 97, 717, 5736, 48340, 422688, 3799080, 34881159, 325750143, 3084634305, 29548452297, 285825135183, 2787990695931, 27391816756281, 270828413410413, 2692692976016352, 26904718314949776, 270017389769189136, 2720718671661444780, 27513054621821846074
Offset: 0

Views

Author

Seiichi Manyama, Dec 09 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=3, s=1, t=3, u=5) = 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)^(2/3)/(1 - x*A(x)^(5/3)) )^3.
G.f. A(x) satisfies A(x) = 1 + x * A(x) * (1 + A(x)^(1/3) + A(x)^(5/3)).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A378883.
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).

A378828 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(2/3)/(1 - x*A(x)^(4/3)) )^3.

Original entry on oeis.org

1, 3, 12, 61, 354, 2220, 14649, 100218, 704373, 5055383, 36895221, 272975652, 2042782905, 15434838759, 117588475377, 902259691317, 6966487019220, 54086849181609, 421986564474946, 3306818224272945, 26015737668878523, 205405810986995869, 1627042895593132485
Offset: 0

Views

Author

Seiichi Manyama, Dec 09 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=3, s=1, t=2, 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) satisfies A(x) = 1/( 1 - x*A(x)^(1/3)/(1 - x*A(x)^(4/3)) )^3.
G.f. A(x) satisfies A(x) = 1 + x * A(x)^(2/3) * (1 + A(x)^(1/3) + A(x)^(5/3)).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A364739.
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.