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.

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.

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.

A349089 a(n) = n! * Sum_{k=0..floor((n-1)/4)} 1 / (4*k+1)!.

Original entry on oeis.org

0, 1, 2, 6, 24, 121, 726, 5082, 40656, 365905, 3659050, 40249550, 482994600, 6278929801, 87905017214, 1318575258210, 21097204131360, 358652470233121, 6455744464196178, 122659144819727382, 2453182896394547640, 51516840824285500441, 1133370498134281009702
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2022

Keywords

Crossrefs

Cf. A002627, A009628, A186763, A334363, A337728, A349088, A352660.

Programs

  • Mathematica
    Table[n! Sum[1/(4 k + 1)!, {k, 0, Floor[(n - 1)/4]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Sin[x] + Sinh[x])/(2 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: (sin(x) + sinh(x)) / (2*(1 - x)).
a(n) = floor(c * n!) for n > 0, where c = 1.008336089... = A334363.
Showing 1-6 of 6 results.

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