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 1-9 of 9 results.

A367135 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^3)).

Original entry on oeis.org

1, 1, 9, 166, 4719, 182326, 8927301, 529922002, 36988772211, 2969132797966, 269488306924833, 27291375956851546, 3050923148547318039, 373187615576953777510, 49580088565083198922845, 7109665420655116517351458, 1094492388113416460752513851
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A000670, A052894, A367134.
Cf. A367137, A367139.

Programs

  • Mathematica
    Table[1/(3*n+1)! * Sum[(3*n+k)! * StirlingS2[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (3*n+k)!*stirling(n, k, 2))/(3*n+1)!;

Formula

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * Stirling2(n,k).
a(n) ~ 3^(4*n) * LambertW(2*exp(1/3)/3)^(3*n + 1) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * 2^(3*n + 1) * exp(n) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367138 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^2)).

Original entry on oeis.org

1, 1, 7, 98, 2096, 60684, 2221766, 98488592, 5129567208, 307066395000, 20775900638472, 1567955813868960, 130596146677118448, 11899839375083061024, 1177540373453616858240, 125754589311488009416704, 14416305655742615673941760, 1765794816084642802179333120
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A007840, A052802, A367139.
Cf. A367134, A367136.

Programs

  • Mathematica
    Table[1/(2*n+1)! * Sum[(2*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (2*n+k)!*abs(stirling(n, k, 1)))/(2*n+1)!;

Formula

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * |Stirling1(n,k)|.
a(n) ~ LambertW(2*exp(3))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(3)))) * exp(n) * (-2 + LambertW(2*exp(3)))^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A376393 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + log(1-x))^3 ).

Original entry on oeis.org

1, 3, 33, 669, 20130, 808902, 40799514, 2480325810, 176637134184, 14428585258896, 1330156753687152, 136632403748954088, 15476220160149512160, 1916493979349783418192, 257601843144279267685056, 37352685483321694825767120, 5812026059839341212943591168, 965974072760231560672817681280
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A367139, A376394.
Cf. A354122.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+log(1-x))^3)/x))
  • PARI
    a(n) = 3*sum(k=0, n, (3*n+k+2)!*abs(stirling(n, k, 1)))/(3*n+3)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 + log(1 - x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A367139.
a(n) = (3/(3*n+3)!) * Sum_{k=0..n} (3*n+k+2)! * |Stirling1(n,k)|.

A367137 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x)^3)).

Original entry on oeis.org

1, 1, 7, 101, 2248, 68024, 2608940, 121316796, 6633841608, 417181294704, 29665022908992, 2353675598751960, 206145540193974288, 19755830347828845360, 2056381966404400741920, 231034314706671715165824, 27865886237401381188422400, 3591366670194210901813749120
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A006252, A198860, A367136.
Cf. A367135, A367139.

Programs

  • Mathematica
    Table[1/(3*n+1)! * Sum[(3*n+k)! * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (3*n+k)!*stirling(n, k, 1))/(3*n+1)!;

Formula

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * Stirling1(n,k).
a(n) ~ LambertW(3*exp(2))^n * n^(n-1) / (sqrt(3*(1 + LambertW(3*exp(2)))) * exp(n) * (3 - LambertW(3*exp(2)))^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A377323 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x)^3)/A(x).

Original entry on oeis.org

1, 1, 5, 53, 884, 20234, 589834, 20903700, 872660256, 41944510752, 2281437791448, 138539360885760, 9290720296262976, 681965664411820944, 54384461861952738528, 4682101594725064872768, 432815761314471190599936, 42757813607285233998385920, 4495579313771176952867958528
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2024

Keywords

Crossrefs

Cf. A365546, A367139, A367152, A377327.
Cf. A365438, A377325.

Programs

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

Formula

a(n) = Sum_{k=0..n} (3*n-k)!/(3*n-2*k+1)! * |Stirling1(n,k)|.

A377327 E.g.f. satisfies A(x) = 1 - A(x)^2 * log(1 - x*A(x)^3).

Original entry on oeis.org

1, 1, 11, 251, 8858, 425534, 25928068, 1916213928, 166580610504, 16657218047328, 1883646389742624, 237695994684785592, 33113333472295201536, 5047818696187818951984, 835818979837614364874496, 149383091745519898076484480, 28663410267058615074689247360, 5877004345535507714104006175616
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2024

Keywords

Crossrefs

Cf. A365546, A367139, A367152, A377323.

Programs

  • PARI
    a(n) = sum(k=0, n, (3*n+2*k)!/(3*n+k+1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (3*n+2*k)!/(3*n+k+1)! * |Stirling1(n,k)|.

A377350 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x)^3)/A(x)^3.

Original entry on oeis.org

1, 1, 1, 11, 108, 1584, 29808, 674988, 18091944, 557844408, 19468760904, 758698622472, 32653135227936, 1538316755200224, 78737559447563136, 4350956519444451840, 258163046132873143680, 16370486288763937324416, 1104824513292622360789248, 79068747951669626322531840
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2024

Keywords

Crossrefs

Cf. A365546, A367139, A367152, A377323, A377327.
Cf. A377325, A377349.

Programs

  • PARI
    a(n) = sum(k=0, (3*n+1)\4, (3*n-3*k)!/(3*n-4*k+1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..floor((3*n+1)/4)} (3*n-3*k)!/(3*n-4*k+1)! * |Stirling1(n,k)|.

A377426 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^4)).

Original entry on oeis.org

1, 1, 11, 254, 9096, 443874, 27487034, 2065181880, 182545878152, 18562391987880, 2134764133508832, 273978733525211472, 38820518588599921200, 6019219063397716575840, 1013766602891962529642832, 184300120562198063868474624, 35971439241165448281366023424
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A007840, A052802, A367138, A367139.
Cf. A377427, A377429.

Programs

  • PARI
    a(n) = sum(k=0, n, (4*n+k)!*abs(stirling(n, k, 1)))/(4*n+1)!;

Formula

a(n) = (1/(4*n+1)!) * Sum_{k=0..n} (4*n+k)! * |Stirling1(n,k)|.

A376394 Expansion of e.g.f. ( (1/x) * Series_Reversion( x*(1 + log(1-x))^3 ) )^(2/3).

Original entry on oeis.org

1, 2, 20, 388, 11382, 449868, 22427988, 1351746912, 95626268208, 7769995319280, 713229439560816, 73000860715645344, 8243857485642410400, 1018250616169754862048, 136561871538665054975520, 19763248903874313555142656, 3069876028020976768409255808, 509447295061343606934940250880
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A367139, A376393.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(1+log(1-x))^3)/x)^(2/3)))
  • PARI
    a(n) = 2*sum(k=0, n, (3*n+k+1)!*abs(stirling(n, k, 1)))/(3*n+2)!;

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367139.
a(n) = (2/(3*n+2)!) * Sum_{k=0..n} (3*n+k+1)! * |Stirling1(n,k)|.
Showing 1-9 of 9 results.

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

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