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.

A366400 G.f. A(x) satisfies A(x) = (1 + x * A(x)^(5/2)) / (1 - x).

Original entry on oeis.org

1, 2, 7, 32, 167, 942, 5593, 34438, 217888, 1407938, 9252168, 61641846, 415412036, 2826736736, 19395080061, 134034296976, 932110471089, 6518146460274, 45805553781349, 323313555424924, 2291130483593189, 16294149468133930, 116259325138469680
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A349311, A366401, A366402, A366403, A366404, A366405, A366406, A366407.
Cf. A262441, A370474.
Cf. A219537, A378890.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k/2, n-k)*binomial(5*k/2, k)/(3*k/2+1));
  • PARI
    a(n, r=2, s=-1, t=4, 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)); \\ Seiichi Manyama, Dec 12 2024

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k/2,n-k) * binomial(5*k/2,k) / (3*k/2+1).
From Seiichi Manyama, Dec 12 2024: (Start)
G.f. A(x) satisfies:
(1) A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^2.
(2) A(x) = 1/( 1 - x*A(x)^(3/2)/(1 + x*A(x)) )^2.
(3) A(x) = 1 + x * A(x) * (1 + A(x)^(3/2)).
(4) A(x) = B(x)^2 where B(x) is the g.f. of A219537.
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). (End)
G.f.: Sum_{k>=0} binomial(5*k/2, k)*x^k/((3*k/2 + 1)*(1 - x)^(5*k/2 + 1)). - Miles Wilson, Feb 02 2025

A378891 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^3.

Original entry on oeis.org

1, 3, 18, 142, 1278, 12429, 127223, 1350456, 14729628, 164079982, 1858781652, 21348787587, 248021665720, 2909439099543, 34413536180688, 409984974779725, 4915119769384221, 59252402698999209, 717819918438472134, 8734481867945979183, 106703642464149880248
Offset: 0

Views

Author

Seiichi Manyama, Dec 10 2024

Keywords

Crossrefs

Cf. A371542, A378889, A378890.
Cf. A378892.

Programs

  • PARI
    a(n, r=3, 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)^(5/3)/(1 + x*A(x)) )^3.
G.f. A(x) satisfies A(x) = 1 + x * A(x) * (1 + A(x)^(4/3) + A(x)^(5/3)).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A378892.
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).
Showing 1-3 of 3 results.

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