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.

A355284 Expansion of e.g.f. 1 / (1 + x + x^2/2 + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 2, 6, 24, 200, 1560, 12936, 130368, 1458432, 17623440, 233922480, 3376625472, 52382131776, 870882440064, 15459372915840, 291596692838400, 5824039155720192, 122814724467223296, 2726547887891407104, 63562453551393223680, 1552499303360183700480
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 26 2022

Keywords

Crossrefs

Cf. A007840, A038205, A102233, A226226, A355285.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + x + x^2/2 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/(1 + x + x^2/2 + log(1 - x)))) \\ Michel Marcus, Jun 27 2022

Formula

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

A365979 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5*k+2) / (5*k+2) ).

Original entry on oeis.org

1, 0, 1, 0, 6, 0, 90, 720, 2520, 51840, 113400, 4276800, 47401200, 444787200, 9725086800, 58378320000, 2029897584000, 30450131712000, 475261239024000, 11952610750080000, 127796530736160000, 4683810971473920000, 90707397988727520000, 1964217505623310080000
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2023

Keywords

Crossrefs

Cf. A226226, A365978.
Cf. A365909.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\5, x^(5*k+2)/(5*k+2)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-2)/5)} (5*k+1)! * binomial(n,5*k+2) * a(n-5*k-2).

A365978 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3*k+2) / (3*k+2) ).

Original entry on oeis.org

1, 0, 1, 0, 6, 24, 90, 1008, 7560, 54432, 712152, 7620480, 81130896, 1266632640, 17587441872, 246734377344, 4527397929600, 77238618702336, 1340945212763520, 28407941067018240, 574640938744314624, 11868502219930137600, 285787326567523173120
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2023

Keywords

Crossrefs

Cf. A226226, A365979.
Cf. A365908.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+2)/(3*k+2)))))

Formula

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

A331559 E.g.f.: -log(1 + x + log(1 - x)).

Original entry on oeis.org

0, 1, 2, 9, 44, 280, 2064, 17738, 172528, 1880856, 22686960, 300193872, 4323063744, 67323469200, 1127433161568, 20205636981840, 385897245967104, 7824675262660608, 167885148101916672, 3800289634376282496, 90513807325761507840, 2262830879094971399424
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 20 2020

Keywords

Comments

Logarithmic transform of A226226.

Crossrefs

Cf. A003713, A032181, A052808, A052832, A226226, A331558.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 + x + Log[1 - x]], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

a(n) ~ (n-1)! / (1 + LambertW(-exp(-2)))^n. - Vaclav Kotesovec, Jan 26 2020

A337060 E.g.f.: 1 / (1 + x^2/2 + log(1 - x)).

Original entry on oeis.org

1, 1, 2, 8, 46, 324, 2708, 26424, 295272, 3714600, 51929472, 798610416, 13399081584, 243556758912, 4767863027328, 100004300847744, 2237419620187776, 53187370914349440, 1338737435337261312, 35568441673932566016, 994744655047298951424, 29211127285363209561600
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2020

Keywords

Crossrefs

Cf. A000266, A007840, A226226, A337061.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 + x^2/2 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 21}]

Formula

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

A337061 E.g.f.: 1 / (1 + x^3/3 + log(1 - x)).

Original entry on oeis.org

1, 1, 3, 12, 72, 534, 4818, 50532, 606408, 8182656, 122712912, 2024328096, 36432644400, 710346495312, 14915647605168, 335567743462944, 8052843408926976, 205328108580310656, 5543345188496499840, 157970863597032124416, 4738694884696030305024
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2020

Keywords

Crossrefs

Cf. A000090, A007840, A226226, A337060.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + x^3/3 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[n_] := a[n] = n (a[n - 1] + (n - 1) a[n - 2]/2) + Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 20}]

Formula

a(0) = a(1) = 1; a(n) = n * (a(n-1) + (n-1) * a(n-2) / 2) + Sum_{k=4..n} binomial(n,k) * (k-1)! * a(n-k).

A338448 E.g.f.: 1 / (1 - x - log(1 - x)).

Original entry on oeis.org

1, 0, -1, -2, 0, 16, 50, -132, -2184, -9984, 6912, 341760, 38544, -47086272, -702019344, -6076389984, -43980940800, -656377887744, -16782743357568, -368775477229824, -6770025717901056, -118247220867640320, -2271088046291742720, -50203882870716579840
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 28 2020

Keywords

Crossrefs

Cf. A006252, A089148, A226226.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - x - Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x + O('x^30)); Vec(serlaplace(1/(1 - x - log(1 - x)))) \\ Michel Marcus, Oct 29 2020

Formula

a(0) = 1; a(n) = -Sum_{k=2..n} binomial(n,k) * (k-1)! * a(n-k).
a(n) ~ -n! / (n * log(n)^2) * (1 - 2*gamma/log(n) + (3*gamma^2 - Pi^2/2)/log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 29 2020

A355285 Expansion of e.g.f. 1 / (1 + x + x^2/2 + x^3/3 + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 6, 24, 120, 720, 7560, 76608, 810432, 9141120, 118015920, 1666336320, 25211774016, 404932155264, 6951992261760, 127203705538560, 2467434718218240, 50477473338494976, 1086707769452699904, 24573149993692615680, 582367494447600583680, 14430857455114783119360
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 26 2022

Keywords

Crossrefs

Cf. A007840, A047865, A226226, A232475, A355284.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 + x + x^2/2 + x^3/3 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/(1 + x + x^2/2 + x^3/3 + log(1 - x)))) \\ Michel Marcus, Jun 27 2022

Formula

E.g.f.: 1 / (1 - Sum_{k>=4} x^k/k).
a(0) = 1; a(n) = Sum_{k=4..n} binomial(n,k) * (k-1)! * a(n-k).
Showing 1-8 of 8 results.

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