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 1-8 of 8 results.

A337725 a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.

Original entry on oeis.org

1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2020

Keywords

Crossrefs

Cf. A000522, A051396, A051397, A087350, A100089, A143820, A330044, A337726, A337727, A337728, A337729, A337730.

Programs

  • Mathematica
    Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 1)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Pi/6 - Sqrt[3] x/2])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 1}], {n, 0, 11}]
Table[Floor[(Exp[3/2] + 2 Sin[(3 Sqrt[3] - Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 1)!], {n, 0, 11}]
  • PARI
    a(n) = (3*n+1)!*sum(k=0, n, 1/(3*k+1)!); \\ Michel Marcus, Sep 17 2020

Formula

E.g.f.: (exp(3*x/2) - 2 * sin(Pi/6 - sqrt(3)*x/2)) / (3*exp(x/2) * (1 - x^3)) = x + 25*x^4/4! + 5251*x^7/7! + 3780721*x^10/10! + ...
a(n) = floor(c * (3*n+1)!), where c = (exp(3/2) + 2 * sin((3 * sqrt(3) - Pi) / 6))/(3 * sqrt(exp(1))) = A143820.

A337727 a(n) = (4*n)! * Sum_{k=0..n} 1 / (4*k)!.

Original entry on oeis.org

1, 25, 42001, 498971881, 21795091762081, 2534333270094778681, 646315807872650838343345, 317599587988620621961919733001, 274101148417699141578015206369183041, 387502275541069630431671657548241448722521, 849931991080760484603611346800010863970028660561
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2020

Keywords

Crossrefs

Cf. A000522, A051396, A051397, A087350, A100733, A330045, A332890, A337725, A337726, A337728, A337729, A337730.

Programs

  • Mathematica
    Table[(4 n)! Sum[1/(4 k)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n)! SeriesCoefficient[(1/2) (Cos[x] + Cosh[x])/(1 - x^4), {x, 0, 4 n}], {n, 0, 10}]
Table[Floor[(1/2) (Cos[1] + Cosh[1]) (4 n)!], {n, 0, 10}]
  • PARI
    a(n) = (4*n)!*sum(k=0, n, 1/(4*k)!); \\ Michel Marcus, Sep 17 2020

Formula

E.g.f.: (1/2) * (cos(x) + cosh(x)) / (1 - x^4) = 1 + 25*x^4/4! + 42001*x^8/8! + 498971881*x^12/12! + ...
a(n) = floor(c * (4*n)!), where c = (cos(1) + cosh(1)) / 2 = A332890.

A337728 a(n) = (4*n+1)! * Sum_{k=0..n} 1 / (4*k+1)!.

Original entry on oeis.org

1, 121, 365905, 6278929801, 358652470233121, 51516840824285500441, 15640512874253077933887601, 8915467710633236496186345872425, 8755702529258688898174686554391144001, 13878488965077362598718732163634314533105081, 33731389859841228248933904149069928786421237268881
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2020

Keywords

Crossrefs

Cf. A000522, A051396, A051397, A087350, A330045, A334363, A337725, A337726, A337727, A337729, A337730.

Programs

  • Mathematica
    Table[(4 n + 1)! Sum[1/(4 k + 1)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 1)! SeriesCoefficient[(1/2) (Sin[x] + Sinh[x])/(1 - x^4), {x, 0, 4 n + 1}], {n, 0, 10}]
Table[Floor[(1/2) (Sin[1] + Sinh[1]) (4 n + 1)!], {n, 0, 10}]
  • PARI
    a(n) = (4*n+1)!*sum(k=0, n, 1/(4*k+1)!); \\ Michel Marcus, Sep 17 2020

Formula

E.g.f.: (1/2) * (sin(x) + sinh(x)) / (1 - x^4) = x + 121*x^5/5! + 365905*x^9/9! + 6278929801*x^13/13! + ...
a(n) = floor(c * (4*n+1)!), where c = (sin(1) + sinh(1)) / 2 = A334363.

A337726 a(n) = (3*n+2)! * Sum_{k=0..n} 1 / (3*k+2)!.

Original entry on oeis.org

1, 61, 20497, 20292031, 44317795705, 180816606476401, 1236785588298582841, 13142083661260741268467, 205016505115667563788085201, 4494781858155895668489979946725, 133764708098719455094261803214536001, 5252940087036713001551661012234828759271
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2020

Keywords

Crossrefs

Cf. A000522, A051396, A051397, A087350, A100043, A143821, A330044, A337725, A337727, A337728, A337729, A337730.

Programs

  • Mathematica
    Table[(3 n + 2)! Sum[1/(3 k + 2)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 2)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Sqrt[3] x/2 + Pi/6])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 2}], {n, 0, 11}]
Table[Floor[(Exp[3/2] - 2 Sin[(3 Sqrt[3] + Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 2)!], {n, 0, 11}]
  • PARI
    a(n) = (3*n+2)!*sum(k=0, n, 1/(3*k+2)!); \\ Michel Marcus, Sep 17 2020

Formula

E.g.f.: (exp(3*x/2) - 2 * sin(sqrt(3)*x/2 + Pi/6)) / (3*exp(x/2) * (1 - x^3)) = x^2/2! + 61*x^5/5! + 20497*x^8/8! + 20292031*x^11/11! + ...
a(n) = floor(c * (3*n+2)!), where c = (exp(3/2) - 2 * sin((3 * sqrt(3) + Pi) / 6))/(3 * sqrt(exp(1))) = A143821.

A337729 a(n) = (4*n+2)! * Sum_{k=0..n} 1 / (4*k+2)!.

Original entry on oeis.org

1, 361, 1819441, 43710250585, 3210080802962401, 563561785768079119561, 202205968733586788098486801, 132994909755454702268136738753721, 148026526435655214537290625514621562305, 262237873172349351865682580536682974917045801, 704454843460345510903820429747302209179158476142321
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2020

Keywords

Crossrefs

Cf. A000522, A051396, A051397, A087350, A330045, A334364, A337725, A337726, A337727, A337728, A337730.

Programs

  • Mathematica
    Table[(4 n + 2)! Sum[1/(4 k + 2)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 2)! SeriesCoefficient[(1/2) (Cosh[x] - Cos[x])/(1 - x^4), {x, 0, 4 n + 2}], {n, 0, 10}]
Table[Floor[(1/2) (Cosh[1] - Cos[1]) (4 n + 2)!], {n, 0, 10}]
  • PARI
    a(n) = (4*n+2)!*sum(k=0, n, 1/(4*k+2)!); \\ Michel Marcus, Sep 17 2020

Formula

E.g.f.: (1/2) * (cosh(x) - cos(x)) / (1 - x^4) = x^2/2! + 361*x^6/6! + 1819441*x^10/10! + 43710250585*x^14/14! + ...
a(n) = floor(c * (4*n+2)!), where c = (cosh(1) - cos(1)) / 2 = A334364.

A337730 a(n) = (4*n+3)! * Sum_{k=0..n} 1 / (4*k+3)!.

Original entry on oeis.org

1, 841, 6660721, 218205219961, 20298322381652065, 4313799472548696853801, 1816972337837511114820981201, 1372104830641374893468212163747161, 1724241814377177346127894133451232399041, 3403694723384093133512770088891935585284510985
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2020

Keywords

Crossrefs

Cf. A000522, A051396, A051397, A087350, A330045, A334365, A337725, A337726, A337727, A337728, A337729.

Programs

  • Mathematica
    Table[(4 n + 3)! Sum[1/(4 k + 3)!, {k, 0, n}], {n, 0, 9}]
Table[(4 n + 3)! SeriesCoefficient[(1/2) (Sinh[x] - Sin[x])/(1 - x^4), {x, 0, 4 n + 3}], {n, 0, 9}]
Table[Floor[(1/2) (Sinh[1] - Sin[1]) (4 n + 3)!], {n, 0, 9}]
  • PARI
    a(n) = (4*n+3)!*sum(k=0, n, 1/(4*k+3)!); \\ Michel Marcus, Sep 17 2020

Formula

E.g.f.: (1/2) * (sinh(x) - sin(x)) / (1 - x^4) = x^3/3! + 841*x^7/7! + 6660721*x^11/11! + 218205219961*x^15/15! + ...
a(n) = floor(c * (4*n+3)!), where c = (sinh(1) - sin(1)) / 2 = A334365.

A352659 a(n) = n! * Sum_{k=0..floor(n/3)} 1 / (3*k)!.

Original entry on oeis.org

1, 1, 2, 7, 28, 140, 841, 5887, 47096, 423865, 4238650, 46625150, 559501801, 7273523413, 101829327782, 1527439916731, 24439038667696, 415463657350832, 7478345832314977, 142088570813984563, 2841771416279691260, 59677199741873516461, 1312898394321217362142
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A000522, A009179, A087350, A143819, A348597, A352660.

Programs

  • Mathematica
    Table[n! Sum[1/(3 k)!, {k, 0, Floor[n/3]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Exp[x] + 2 Exp[-x/2] Cos[Sqrt[3] x/2])/(3 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    a(n) = n! * sum(k=0, n\3, 1/(3*k)!); \\ Michel Marcus, Mar 29 2022

Formula

E.g.f.: (exp(x) + 2 * exp(-x/2) * cos(sqrt(3)*x/2)) / (3*(1 - x)).
a(n) = floor(c * n!), where c = 1.16805831... = A143819.

A119831 Number triangle (3n)!/(3k)!.

Original entry on oeis.org

1, 6, 1, 720, 120, 1, 362880, 60480, 504, 1, 479001600, 79833600, 665280, 1320, 1, 1307674368000, 217945728000, 1816214400, 3603600, 2730, 1, 6402373705728000, 1067062284288000, 8892185702400, 17643225600, 13366080, 4896, 1
Offset: 0

Views

Author

Paul Barry, May 25 2006

Keywords

Comments

Row sums are A087350. Inverse is bi-diagonal array A119832.

Examples

			Triangle begins
1,
6, 1,
720, 120, 1,
362880, 60480, 504, 1,
479001600, 79833600, 665280, 1320, 1,
1307674368000, 217945728000, 1816214400, 3603600, 2730, 1

Crossrefs

Cf. A119828.

Programs

  • Mathematica
    Flatten[Table[(3n)!/(3k)!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jan 19 2013 *)

Formula

Number triangle T(n,k)=[k<=n]*(3n)!/(3k)!
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.