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 1-6 of 6 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)

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

A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

Programs

  • PARI
    a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));

Formula

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A370056 a(n) = 2*(4*n+1)!/(3*n+2)!.

Original entry on oeis.org

1, 2, 18, 312, 8160, 287280, 12751200, 684028800, 43062243840, 3113350732800, 254265345734400, 23153103246873600, 2326025084653670400, 255579097716214272000, 30491180727539051520000, 3925248256199788277760000, 542357159056633603178496000
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2024

Keywords

Crossrefs

Cf. A365340, A370057, A370058.
Cf. A000142, A069271.

Programs

  • PARI
    a(n) = 2*(4*n+1)!/(3*n+2)!;

Formula

E.g.f.: exp( 1/2 * Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A069271(n).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*a(n) -8*(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)^(3/2))^2.
a(n) = 2 * Sum_{k=0..n} (3*n+2)^(k-1) * |Stirling1(n,k)|. (End)

A375871 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)) - 1) ).

Original entry on oeis.org

1, 3, 30, 543, 14493, 515001, 22930869, 1229340027, 77151412902, 5551075890453, 450607640485269, 40745592546015495, 4061982705195354033, 442649982865922396337, 52351468801767526253538, 6678605910447082873015923, 914198409310749883430655441
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Cf. A349683, A375870, A375872.
Cf. A027710, A370057.

Programs

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

Formula

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

A380946 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(-3*x) ).

Original entry on oeis.org

1, 6, 111, 3678, 179073, 11588688, 938905551, 91542271824, 10444685410881, 1365936450693120, 201503447217869679, 33108736185915906816, 5997057218957213126721, 1187319940110958086623232, 255104922613608981003351375, 59120580081196768991316314112
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2025

Keywords

Links

Crossrefs

Cf. A370057, A377833.
Cf. A380647, A380724.

Programs

  • PARI
    a(n, q=3, r=3, s=3, t=0, u=1) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(3*x*A(x))/(1 - x*A(x))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380724.
a(n) = 3 * n! * Sum_{k=0..n} (3*n+3)^(k-1) * binomial(4*n-k+2,n-k)/k!.
Showing 1-6 of 6 results.

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