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.

A367134 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^2)).

Original entry on oeis.org

1, 1, 7, 97, 2051, 58681, 2122695, 92960001, 4782826459, 282821367001, 18901822316543, 1409070858589153, 115925274671836371, 10433564954705754681, 1019782291631652745591, 107570331041074850633473, 12180277895590328004331019, 1473587743517654702900335705
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A000670, A052894, A367135.
Cf. A367136, A367138.

Programs

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

Formula

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

A367139 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^3)).

Original entry on oeis.org

1, 1, 9, 167, 4780, 186004, 9173780, 548563140, 38573633016, 3119384230176, 285237426927552, 29102185296785160, 3277703460197645232, 403931173342682581296, 54066960915411480743520, 7811249803193620134996864, 1211525560869437165319590400
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A007840, A052802, A367138.
Cf. A367135, A367137.

Programs

  • Mathematica
    Table[1/(3*n+1)! * Sum[(3*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (3*n+k)!*abs(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(4))^n * n^(n-1) / (sqrt(3*(1 + LambertW(3*exp(4)))) * exp(n) * (-3 + LambertW(3*exp(4)))^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

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

A376390 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^3 ).

Original entry on oeis.org

1, 3, 33, 666, 19923, 795438, 39849549, 2405748978, 170114699247, 13796351753670, 1262691211748865, 128760309960844554, 14478116911623185163, 1779761344294187865198, 237465809999666515842261, 34179385495053448088261154, 5279029838285444642785757415, 870905593631158913782753290198
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A367135, A376391.
Cf. A226515.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = 1/(2 - exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A367135.
a(n) = (3/(3*n+3)!) * Sum_{k=0..n} (3*n+k+2)! * Stirling2(n,k).

A377324 E.g.f. satisfies A(x) = 1 + (exp(x*A(x)^3) - 1)/A(x).

Original entry on oeis.org

1, 1, 5, 52, 839, 18436, 513797, 17366224, 690366875, 31565619916, 1632064968929, 94159057903384, 5996889060457055, 417920884113926740, 31634205840603000221, 2584579552124805784672, 226699825143636127509347, 21247444370267806167804316, 2119206766514801966851437113
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2024

Keywords

Crossrefs

Cf. A052752, A367135, A367181, A377328.
Cf. A367180, A377326.

Programs

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

Formula

a(n) = Sum_{k=0..n} (3*n-k)!/(3*n-2*k+1)! * Stirling2(n,k).

A377328 E.g.f. satisfies A(x) = 1 + A(x)^2 * (exp(x*A(x)^3) - 1).

Original entry on oeis.org

1, 1, 11, 250, 8789, 420646, 25536083, 1880370598, 162872596937, 16227667154806, 1828467483194975, 229904271890603014, 31913005486577248877, 4847412341607090455110, 799762918909215143560907, 142427688272456020835132518, 27231132645610171996487568017, 5563389652463220933157357670806
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2024

Keywords

Crossrefs

Cf. A052752, A367135, A367181, A377324.

Programs

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

Formula

a(n) = Sum_{k=0..n} (3*n+2*k)!/(3*n+k+1)! * Stirling2(n,k).

A377348 E.g.f. satisfies A(x) = 1 + (exp(x*A(x)^3) - 1)/A(x)^3.

Original entry on oeis.org

1, 1, 1, 10, 79, 946, 14653, 267478, 5817187, 145061146, 4089128425, 128703410254, 4470302200087, 169912192575490, 7014628977829237, 312570024564324358, 14952747796689292747, 764341021646724256426, 41578052013117358139809, 2398149800670737138081470
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2024

Keywords

Crossrefs

Cf. A052752, A367135, A367181, A377324, A377328.
Cf. A377326, A377347.

Programs

  • PARI
    a(n) = sum(k=0, (3*n+1)\4, (3*n-3*k)!/(3*n-4*k+1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..floor((3*n+1)/4)} (3*n-3*k)!/(3*n-4*k+1)! * Stirling2(n,k).

A377424 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^4)).

Original entry on oeis.org

1, 1, 11, 253, 9019, 438021, 26992707, 2018069341, 177498369419, 17959376607061, 2055112480694323, 262437681414074541, 36999068388057870651, 5708040382071000644581, 956533539112835413864739, 173022072326584494697760893, 33600521994423195247370822251, 6972639514725247888782370422261
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A000670, A052894, A367134, A367135.
Cf. A377425, A377428.

Programs

  • PARI
    a(n) = sum(k=0, n, (4*n+k)!*stirling(n, k, 2))/(4*n+1)!;

Formula

a(n) = (1/(4*n+1)!) * Sum_{k=0..n} (4*n+k)! * Stirling2(n,k).

A376391 Expansion of e.g.f. ( (1/x) * Series_Reversion( x*(2 - exp(x))^3 ) )^(2/3).

Original entry on oeis.org

1, 2, 20, 386, 11252, 441722, 21867764, 1308580226, 91904288420, 7413237414602, 675503178005108, 68631619821747842, 7693344955213551428, 943236099444038389082, 125565496331888560573172, 18037220418654308659836674, 2780985275750966018759898212, 458079154394191702424821932842
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A367135, A376390.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367135.
a(n) = (2/(3*n+2)!) * Sum_{k=0..n} (3*n+k+1)! * Stirling2(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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