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.

A378892 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 5, 37, 322, 3067, 30951, 325171, 3519038, 38959997, 439177850, 5023590609, 58163050071, 680308820750, 8026782091957, 95419476630100, 1141762194395927, 13740910664096101, 166216043531507231, 2019807368837970964, 24644779751103948475, 301818330734940817283
Offset: 0

Views

Author

Seiichi Manyama, Dec 10 2024

Keywords

Crossrefs

Cf. A001764, A271469, A363982, A364736, A364864.
Cf. A378891.

Programs

  • PARI
    a(n, r=1, s=-1, t=6, u=3) = 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

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^5/(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(s*k,n-k)/(t*k+u*(n-k)+r).

A378889 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(4/3)/(1 + x*A(x)^(1/3)) )^3.

Original entry on oeis.org

1, 3, 12, 61, 348, 2127, 13617, 90132, 611802, 4235405, 29788821, 212255520, 1528928674, 11115361491, 81452537253, 601004875689, 4461440570523, 33295962947925, 249673885001674, 1880204670772221, 14213624028779964, 107823953314047139, 820541644515512502
Offset: 0

Views

Author

Seiichi Manyama, Dec 10 2024

Keywords

Crossrefs

Cf. A371542, A378890, A378891.
Cf. A364758.

Programs

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

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)/(1 + x*A(x)^(1/3)) )^3.
G.f. A(x) satisfies A(x) = 1 + x * A(x)^(1/3) * (1 + A(x)^(4/3) + A(x)^(5/3)).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A364758.
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(s*k,n-k)/(t*k+u*(n-k)+r).

A378890 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(5/3)/(1 + x*A(x)^(2/3)) )^3.

Original entry on oeis.org

1, 3, 15, 97, 711, 5613, 46552, 399918, 3527553, 31761600, 290721387, 2697131541, 25304974597, 239684681523, 2288849098119, 22012319667437, 213011739042714, 2072597720747352, 20264567643461700, 198998140737895692, 1961831436443431818, 19409477239837165874
Offset: 0

Views

Author

Seiichi Manyama, Dec 10 2024

Keywords

Crossrefs

Cf. A371542, A378889, A378891.
Cf. A365225.

Programs

  • PARI
    a(n, r=3, s=-1, t=5, u=2) = 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

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^(4/3)/(1 + x*A(x)^(2/3)) )^3.
G.f. A(x) satisfies A(x) = 1 + x * A(x)^(2/3) * (1 + A(x)^(4/3) + A(x)^(5/3)).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A365225.
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(s*k,n-k)/(t*k+u*(n-k)+r).

A378952 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)^(4/3)) )^3.

Original entry on oeis.org

1, 3, 18, 139, 1218, 11511, 114398, 1178421, 12469626, 134734092, 1480317468, 16487870031, 185744716414, 2112756042468, 24230663513604, 279889210974003, 3253295301115290, 38023971948455859, 446603044829013514, 5268557500949993964, 62398899992129490756
Offset: 0

Views

Author

Seiichi Manyama, Dec 11 2024

Keywords

Crossrefs

Cf. A371724, A378951.
Cf. A364765, A378954.
Cf. A378891.

Programs

  • PARI
    a(n, r=3, s=-1, t=6, 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

G.f. A(x) satisfies:
(1) A(x) = 1/( 1 - x*A(x)^(5/3)/(1 + x*A(x)^(4/3)) )^3.
(2) A(x) = 1 + x * A(x)^(4/3) * (1 + A(x)^(2/3) + A(x)^(4/3)).
(3) A(x) = B(x)^3 where B(x) is the g.f. of A364765.
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(s*k,n-k)/(t*k+u*(n-k)+r).

A378951 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(5/3)/(1 + x*A(x)) )^3.

Original entry on oeis.org

1, 3, 15, 94, 663, 5025, 39970, 329145, 2782095, 23999078, 210427869, 1869908364, 16802935370, 152425394958, 1393972037301, 12838326815582, 118970843349711, 1108503805898190, 10378559702646846, 97593299922016224, 921294705307189029
Offset: 0

Views

Author

Seiichi Manyama, Dec 11 2024

Keywords

Crossrefs

Cf. A371724, A378952.
Cf. A271469, A370474.
Cf. A378891.

Programs

  • PARI
    a(n, r=3, s=-1, t=5, u=3) = 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

G.f. A(x) satisfies:
(1) A(x) = 1/( 1 - x*A(x)^(4/3)/(1 + x*A(x)) )^3.
(2) A(x) = 1 + x * A(x) * (1 + A(x)^(2/3) + A(x)^(4/3)).
(3) A(x) = B(x)^3 where B(x) is the g.f. of A271469.
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(s*k,n-k)/(t*k+u*(n-k)+r).
Showing 1-5 of 5 results.

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