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 1-10 of 11 results. Next

A365977 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5*k+1) / (5*k+1) ).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 840, 6720, 60480, 604800, 6652800, 83462400, 1138233600, 16746912000, 264176640000, 4444771968000, 80719172352000, 1556132497920000, 31722198842880000, 681437830993920000, 15378172899747840000, 366025806545817600000
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2023

Keywords

Crossrefs

Cf. A296676, A365975, A365976.
Cf. A365969.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\5, x^(5*k+1)/(5*k+1)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} (5*k)! * binomial(n,5*k+1) * a(n-5*k-1).

A331618 E.g.f.: exp(1 / (1 - arctanh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 15, 97, 785, 7523, 83615, 1053281, 14838177, 230832867, 3929944623, 72633052545, 1447981700529, 30960823851267, 706676217730239, 17145815895371073, 440594781536265537, 11952178787661839427, 341291300477569866831, 10231558345117929439521
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Crossrefs

Cf. A000246, A080833, A202139, A296676, A331610, A331615, A331616, A331617.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcTanh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A296676[0] = 1; A296676[n_] := A296676[n] = Sum[Binomial[n, k] If[OddQ[k], (k - 1)!, 0] A296676[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A296676[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - atanh(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A296676(k) * a(n-k).
a(n) ~ (exp(2) + 1)^(n - 1/4) * n^(n - 1/4) / ((exp(2) - 1)^(n + 1/4) * exp(n - 4*exp(1)*sqrt(n/(exp(4) - 1)) - 2/(exp(4) - 1) - 1/2)). - Vaclav Kotesovec, Jan 26 2020

A365975 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3*k+1) / (3*k+1) ).

Original entry on oeis.org

1, 1, 2, 6, 30, 180, 1260, 10800, 104760, 1130760, 13776480, 184044960, 2670220080, 42222280320, 718144004160, 13061603808000, 254036916144000, 5247117638294400, 114652672773408000, 2647321293055507200, 64330669872690566400, 1640738743703289331200
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2023

Keywords

Crossrefs

Cf. A296676, A365976, A365977.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+1)/(3*k+1)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} (3*k)! * binomial(n,3*k+1) * a(n-3*k-1).

A365976 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4*k+1) / (4*k+1) ).

Original entry on oeis.org

1, 1, 2, 6, 24, 144, 1008, 8064, 72576, 766080, 8934912, 113895936, 1573254144, 23864924160, 389247344640, 6786673496064, 125855767166976, 2492616008171520, 52243870155079680, 1154797100239749120, 26835102086208159744, 656159502089355264000
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2023

Keywords

Crossrefs

Cf. A296676, A365975, A365977.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\4, x^(4*k+1)/(4*k+1)))))

Formula

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

A365980 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2*k+3) / (2*k+3) ).

Original entry on oeis.org

1, 0, 0, 2, 0, 24, 80, 720, 5376, 53760, 490752, 6289920, 68766720, 1024607232, 13520332800, 226177695744, 3498759290880, 65257155624960, 1153246338220032, 23793010526453760, 472374431008948224, 10686755493583257600, 235406405307208826880
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2023

Keywords

Crossrefs

Cf. A355284, A365981, A365982.
Cf. A130915, A296676.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x-atanh(x))))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/2)} (2*k+2)! * binomial(n,2*k+3) * a(n-2*k-3).
E.g.f.: 1 / ( 1 + x - arctanh(x) ).

A385468 Expansion of e.g.f. 1/(1 - 2 * arctanh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 17, 129, 1269, 15147, 213765, 3475329, 64020585, 1317472563, 29960707545, 746086414785, 20192521440285, 590166330458715, 18525204423695565, 621571306435103745, 22199954036873457105, 840913892465144800995, 33672216851574639900705
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2025

Keywords

Crossrefs

Cf. A296676, A385469.
Cf. A001147, A111594, A385376.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*atanh(x))^(1/2)))

Formula

E.g.f.: 1/(1 - log((1+x)/(1-x)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * A111594(n,k).
a(n) ~ 2 * (exp(1) + 1)^(n - 1/2) * n^n / (exp(n - 1/2) * (exp(1) - 1)^(n + 1/2)). - Vaclav Kotesovec, Jun 30 2025

A385469 Expansion of e.g.f. 1/(1 - 3 * arctanh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 30, 312, 4224, 70176, 1384032, 31590912, 819254016, 23792039424, 764912590848, 26970073390080, 1034798724320256, 42921327875788800, 1913760046417508352, 91281373260924026880, 4637755280044146032640, 250054580636566927441920, 14259891701316651909120000
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2025

Keywords

Crossrefs

Cf. A296676, A385468.
Cf. A007559, A111594.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*atanh(x))^(1/3)))

Formula

E.g.f.: 1/(1 - (3/2) * log((1+x)/(1-x)))^(1/3).
a(n) = Sum_{k=0..n} A007559(k) * A111594(n,k).

A385470 Expansion of e.g.f. 1/(1 - 2 * arctanh(x)).

Original entry on oeis.org

1, 2, 8, 52, 448, 4848, 62912, 952992, 16496640, 321282816, 6952332288, 165489858048, 4297340166144, 120890184308736, 3662409013420032, 118879239686541312, 4115985952586858496, 151415632063102648320, 5897814669785134006272, 242489327746828076974080
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2025

Keywords

Crossrefs

Cf. A296676, A385471.
Cf. A111594, A385468.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-2ArcTanh[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 04 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*atanh(x))))

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A385468.
E.g.f.: 1/(1 - log((1+x)/(1-x))).
a(n) = Sum_{k=0..n} 2^k * k! * A111594(n,k).
a(n) ~ 2^(3/2) * sqrt(Pi) * (1 + exp(1))^(n-1) * n^(n + 1/2) / (exp(n-1) * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Jun 30 2025

A385471 Expansion of e.g.f. 1/(1 - 3 * arctanh(x)).

Original entry on oeis.org

1, 3, 18, 168, 2088, 32472, 605952, 13192848, 328268160, 9189103104, 285808290048, 9778434400512, 364965976571904, 14756982055363584, 642580290860378112, 29979230177385750528, 1491908801018949697536, 78884742832151951278080, 4416389166601900315901952
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2025

Keywords

Crossrefs

Cf. A296676, A385470.
Cf. A111594, A385469.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*atanh(x))))

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A385469.
E.g.f.: 1/(1 - (3/2) * log((1+x)/(1-x))).
a(n) = Sum_{k=0..n} 3^k * k! * A111594(n,k).

A385472 Expansion of e.g.f. 1/(1 - arctanh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 23, 201, 2529, 36027, 633975, 12445521, 282376065, 7045758003, 196111046295, 5929900611225, 195773173735905, 6950809317622635, 265652001656970615, 10828342476187312545, 470368564694268015105, 21643209863062015977315, 1053344875062427351601175
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2025

Keywords

Crossrefs

Cf. A296676, A385473.
Cf. A001147, A111594.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-atanh(2*x))^(1/2)))

Formula

E.g.f.: 1/(1 - (1/2) * log((1+2*x)/(1-2*x)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A111594(n,k).
Showing 1-10 of 11 results. Next

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