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.

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).

A372165 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(5/2) / (1 - x * A(x)) ).

Original entry on oeis.org

1, 2, 28, 758, 31160, 1730562, 121434364, 10312487054, 1028675082960, 117917384790914, 15275849114906804, 2207219937751153998, 351952462602081499480, 61392924661901606654402, 11629541557015551899838252, 2377438129669444985664704078, 521710054052646408966825988256
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2024

Keywords

Crossrefs

Cf. A371581, A372164, A372183.

Programs

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

Formula

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A372183.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.

A371582 G.f. satisfies A(x) = ( 1 + x*A(x)^3 / (1 - x*A(x)) )^2.

Original entry on oeis.org

1, 2, 15, 146, 1623, 19526, 247516, 3256118, 44037023, 608484766, 8552832116, 121908218724, 1757915510695, 25598937436696, 375916184707142, 5560517754432772, 82774606577536376, 1239110145377709862, 18641533742708676711, 281697878640036748684
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2024

Keywords

Crossrefs

Cf. A006013, A211789, A365146, A371581.
Cf. A371575.

Programs

  • PARI
    a(n, r=2, s=1, t=6, 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

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.

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