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.

A365340 a(n) = (4*n)!/(3*n+1)!.

Original entry on oeis.org

1, 1, 8, 132, 3360, 116280, 5100480, 271252800, 16963914240, 1220096908800, 99225500774400, 9003984596006400, 901928094049382400, 98856066097780992000, 11768525894839633920000, 1512185803617951221760000, 208598907329474462760960000
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2023

Keywords

Crossrefs

Cf. A001761, A001763, A052795, A365341.
Cf. A004331, A061924.
Cf. A370056, A370057, A370058.
Cf. A000142, A002293.

Programs

  • PARI
    a(n) = (4*n)!/(3*n+1)!;
  • Python
    from sympy import ff
def A365340(n): return ff(n<<2,n-1) # Chai Wah Wu, Sep 01 2023

Formula

E.g.f.: exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
a(n) = A000142(n)*A002293(n). - Alois P. Heinz, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^3).
a(n) = Sum_{k=0..n} (3*n+1)^(k-1) * |Stirling1(n,k)|. (End)

A370057 a(n) = 3*(4*n+2)!/(3*n+3)!.

Original entry on oeis.org

1, 3, 30, 546, 14688, 526680, 23680800, 1282554000, 81339793920, 5915366392320, 485415660038400, 44376781223174400, 4473125162795520000, 492902545595556096000, 58949616073242166272000, 7605168496387089788160000, 1052810955815818170875904000
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2024

Keywords

Links

Crossrefs

Cf. A365340, A370056, A370058.
Cf. A000142, A006632.

Programs

  • Mathematica
    Table[(3(4n+2)!)/(3n+3)!,{n,0,20}] (* Harvey P. Dale, Feb 15 2025 *)
  • PARI
    a(n) = 3*(4*n+2)!/(3*n+3)!;

Formula

E.g.f.: exp( 3/4 * Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A006632(n+1).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) -8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x))^3.
a(n) = 3 * Sum_{k=0..n} (3*n+3)^(k-1) * |Stirling1(n,k)|. (End)

A370058 a(n) = 4*(4*n+3)!/(3*n+4)!.

Original entry on oeis.org

1, 4, 44, 840, 23256, 850080, 38750400, 2120489280, 135566323200, 9922550077440, 818544054182400, 75160674504115200, 7604312776752384000, 840608992488545280000, 100812386907863414784000, 13037431708092153922560000, 1808675231786149165350912000
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2024

Keywords

Crossrefs

Cf. A365340, A370056, A370057.
Cf. A065866, A370055.
Cf. A000142, A002293.

Programs

  • PARI
    a(n) = 4*(4*n+3)!/(3*n+4)!;

Formula

E.g.f.: exp( Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A002293(n+1).
D-finite with recurrence 3*(3*n+2)*(3*n+4)*(n+1)*a(n) -8*n*(4*n+1)*(2*n+1)*(4*n+3)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^(3/4))^4.
a(n) = 4 * Sum_{k=0..n} (3*n+4)^(k-1) * |Stirling1(n,k)|. (End)
E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^4 ). - Seiichi Manyama, Feb 06 2025

A375870 E.g.f. satisfies A(x) = exp( 2 * (exp(x*A(x)^(3/2)) - 1) ).

Original entry on oeis.org

1, 2, 18, 310, 8038, 280264, 12313242, 653591922, 40704551630, 2910862397646, 235114931752898, 21172206066055312, 2103333121459719446, 228525476912967164714, 26957670075375556803178, 3431314158743477432894790, 468762478424957403561956702
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Cf. A349683, A375871, A375872.
Cf. A001861, A370056.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (3*n+2)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A349683.
a(n) = 2 * Sum_{k=0..n} (3*n+2)^(k-1) * Stirling2(n,k).
Showing 1-4 of 4 results.

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