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 1-7 of 7 results.

A346921 Expansion of e.g.f. 1 / (1 - log(1 - x)^2 / 2).

Original entry on oeis.org

1, 0, 1, 3, 17, 110, 874, 8064, 85182, 1012248, 13369026, 194245590, 3079135806, 52880064588, 978038495316, 19381794788160, 409702099828104, 9201877089355584, 218832476773294008, 5493266481129425064, 145153549897858762776, 4027310838211114515600
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346922, A346923, A346924.
Cf. A000254, A052811, A305306, A330047, A347001.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - Log[1 - x]^2/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^2/2))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/2^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,2)| * a(n-k).
a(n) ~ n! * exp(sqrt(2)*n) / (sqrt(2) * (exp(sqrt(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/2^k. - Seiichi Manyama, May 06 2022

A347001 Expansion of e.g.f. exp( log(1 - x)^2 / 2 ).

Original entry on oeis.org

1, 0, 1, 3, 14, 80, 544, 4284, 38310, 383256, 4239006, 51345690, 675770028, 9600349824, 146396925648, 2384700728760, 41320373582652, 758780222426592, 14718569154071964, 300706641183038292, 6453691377726073128, 145154958710291611200, 3414131149418742544320
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A000254, A060311, A305306, A346921, A347002, A347003, A347004.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/(2^k*k!)); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,2)| * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/(2^k * k!). - Seiichi Manyama, May 06 2022

A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).

Original entry on oeis.org

1, 1, 3, 17, 120, 1084, 11642, 146446, 2101656, 33958344, 609431232, 12033015840, 259163792016, 6047213451408, 151953760489008, 4091057804809104, 117485988199385088, 3584814699783432960, 115816462543697120640, 3949619921174717629056, 141780511159572486530304, 5344008726418981985707776
Offset: 0

Views

Author

Ilya Gutkovskiy, May 29 2018

Keywords

Comments

a(n)/n! is the invert transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1084*x^5/5! + 11642*x^6/6! + ...

Links

Crossrefs

Cf. A006252, A024167, A058312, A058313, A305306.

Programs

  • Maple
    g:= proc(n) g(n):= `if`(n=1, 0, g(n-1))-(-1)^n/n end:
b:= proc(n) option remember; `if`(n=0, 1,
      add(g(j)*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - Log[1 + x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/(1 - Sum[Sum[(-1)^(j + 1)/j, {j, 1, k}] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[((-1)^(k + 1) LerchPhi[-1, 1, k + 1] + Log[2]) a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 21}]

Formula

E.g.f.: 1/(1 - Sum_{k>=1} (A058313(k)/A058312(k))*x^k).
a(n) ~ n! * (2 - LambertW(exp(2))) / ((1 + 1/LambertW(exp(2))) * (LambertW(exp(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021

A128044 a(n) = numerator of b(n), where b(1) = 1, b(n) = Sum_{k=1..n-1} b(n-k) * H(k); H(k) = Sum_{j=1..k} 1/j, the k-th harmonic number.

Original entry on oeis.org

1, 1, 5, 35, 27, 156, 25951, 419681, 646379, 13439609, 5544403, 56359019, 109370096651, 218981057573, 1073115579569, 334684898286103, 8505202310547841, 515483074900523, 712333151156230489
Offset: 1

Views

Author

Leroy Quet, Feb 11 2007

Keywords

Examples

			1, 1, 5/2, 35/6, 27/2, 156/5, 25951/360, 419681/2520, 646379/1680, 13439609/15120, 5544403/2700, 56359019/11880, ...

Crossrefs

Cf. A001008, A002805, A128045 (denominators), A305306.

Programs

  • Mathematica
    f[l_List] := Block[{n = Length[l] + 1},Append[l, Sum[l[[n - k]]*HarmonicNumber[k], {k, n - 1}]]];Numerator[Nest[f, {1}, 20]] (* Ray Chandler, Feb 12 2007 *)

Formula

G.f. for fractions: x / (1 + log(1 - x) / (1 - x)). - Ilya Gutkovskiy, Sep 01 2021

Extensions

Extended by Ray Chandler, Feb 12 2007

A128045 a(n) = denominator of b(n), where b(1) = 1, b(n) = Sum_{k=1..n-1} b(n-k) * H(k); H(k) = Sum_{j=1..k} 1/j, the k-th harmonic number.

Original entry on oeis.org

1, 1, 2, 6, 2, 5, 360, 2520, 1680, 15120, 2700, 11880, 9979200, 8648640, 18345600, 2476656000, 27243216000, 714714000, 427508928000, 1160381376000, 1055947052160000, 22174888095360000, 38718058579200, 141031842336000
Offset: 1

Views

Author

Leroy Quet, Feb 11 2007

Keywords

Examples

			1, 1, 5/2, 35/6, 27/2, 156/5, 25951/360, 419681/2520, 646379/1680, 13439609/15120, 5544403/2700, 56359019/11880, ...

Crossrefs

Cf. A001008, A002805, A128044 (numerators), A305306.

Programs

  • Mathematica
    f[l_List] := Block[{n = Length[l] + 1},Append[l, Sum[l[[n - k]]*HarmonicNumber[k], {k, n - 1}]]];Denominator[Nest[f, {1}, 24]] (* Ray Chandler, Feb 12 2007 *)

Formula

G.f. for fractions: x / (1 + log(1 - x) / (1 - x)). - Ilya Gutkovskiy, Sep 01 2021

Extensions

Extended by Ray Chandler, Feb 12 2007

A362911 Expansion of e.g.f. 1/( 1 - (1 + x) * log(1 + x) ).

Original entry on oeis.org

1, 1, 3, 11, 60, 384, 3062, 27838, 293416, 3447768, 45277392, 651587760, 10254900048, 174557518992, 3203361670896, 62938642659504, 1319693558377728, 29390794198726656, 693223221342879360, 17256288944072200320, 452215395177034040064
Offset: 0

Views

Author

Seiichi Manyama, May 10 2023

Keywords

Crossrefs

Cf. A305306, A362912.
Cf. A005727, A006153, A305307.

Programs

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

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * A006153(k).
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / ((1 + LambertW(1)) * exp(n) * (1/LambertW(1) - 1)^(n+1)). - Vaclav Kotesovec, Nov 11 2023

A362912 Expansion of e.g.f. 1/( 1 - (exp(x) - 1) * exp(exp(x) - 1) ).

Original entry on oeis.org

1, 1, 5, 34, 303, 3371, 45016, 701401, 12490057, 250215916, 5569582777, 136371309999, 3642603629462, 105405416033607, 3284722016179597, 109672448519030698, 3905936524326557659, 147802493781420536423, 5921911678533323178312
Offset: 0

Views

Author

Seiichi Manyama, May 10 2023

Keywords

Crossrefs

Cf. A305306, A362911.
Cf. A006153, A347339.

Programs

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

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A006153(k).
a(n) ~ n! * LambertW(1) / ((1 + LambertW(1))^2 * (log(1 + LambertW(1)))^(n+1)). - Vaclav Kotesovec, Nov 11 2023
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.