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.

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A302548 Expansion of e.g.f. -log(1 + log(1 - x))/(1 + log(1 - x)).

Original entry on oeis.org

0, 1, 4, 22, 155, 1333, 13541, 158688, 2107682, 31291894, 513590170, 9234669420, 180534475832, 3812852144788, 86517295628188, 2099170738243328, 54233876338638192, 1486517654443664016, 43084555863325589232, 1316588795487600071904, 42306543064537291007424, 1426115146736949130634400
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 22*x^3/3! + 155*x^4/4! + 1333*x^5/5! + 13541*x^6/6! + ...

Links

Crossrefs

Cf. A000254, A001008, A002805, A003713, A007840, A073596, A222058, A300490, A302547.

Programs

  • Maple
    H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*H(k)*k!, k=1..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[-Log[1 + Log[1 - x]]/(1 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 21}]

Formula

a(n) = Sum_{k=1..n} |Stirling1(n,k)|*H(k)*k!, where H(k) is the k-th harmonic number.
a(n) ~ sqrt(2*Pi) * log(n) * n^(n + 1/2) / (exp(1)-1)^(n+1). - Vaclav Kotesovec, Jun 23 2018

A346946 Expansion of e.g.f. log( 1 + log(1 + x)^4 / 4! ).

Original entry on oeis.org

1, -10, 85, -735, 6734, -66024, 693230, -7774250, 92759821, -1172483598, 15630569591, -218793782025, 3201481037819, -48746860400024, 768683653934928, -12487871805640344, 207761719406853466, -3513910668343842900, 59833161662103132050, -1011244718827893629750
Offset: 4

Views

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

Crossrefs

Cf. A000454, A003713, A346944, A346945, A346947.
Cf. A346976.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS1[n, 4] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 23}]

Formula

a(n) = Stirling1(n,4) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,4) * k * a(k).
a(n) = Sum_{k=1..floor(n/4)} (-1)^(k-1) * (4*k)! * Stirling1(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

Original entry on oeis.org

1, 2, 15, 234, 6170, 245755, 13761937, 1030431500, 99399019626, 12003835242090, 1773907219147800, 314880916127332489, 66109411013740671200, 16204039283106534720952, 4585484528618722750937783, 1483746673734716952089913364, 544359300175753347889146067840
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Examples

			The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))

Links

Crossrefs

Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.

Programs

  • Maple
    g:= x-> -log(1-x):
a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022
  • Mathematica
    Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]
  • PARI
    T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, n); \\ Seiichi Manyama, Feb 11 2022

Formula

a(n) = T(n,n), T(n,k) = Sum_{j=1..n} |Stirling1(n,j)| * T(j,k-1), k>1, T(n,1) = (n-1)!. - Seiichi Manyama, Feb 11 2022

A305323 Expansion of e.g.f. 1/(1 + log(1 + log(1 - x))).

Original entry on oeis.org

1, 1, 4, 25, 211, 2238, 28560, 425808, 7261200, 139367278, 2973006344, 69775267186, 1786673529746, 49565881948204, 1480900541242572, 47407364553205448, 1618838460981098680, 58734896900587841824, 2256402484187691207152, 91499934912942249975504, 3905739517580787866827872
Offset: 0

Views

Author

Ilya Gutkovskiy, May 30 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 25*x^3/3! + 211*x^4/4! + 2238*x^5/5! + 28560*x^6/6! + ...

Links

Crossrefs

Cf. A003713, A007840, A217033, A305988.

Programs

  • Maple
    S:= series(1/(1+log(1+log(1-x))),x,31):
seq(coeff(S,x,n)*n!,n=0..30); # Robert Israel, May 31 2018
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + Log[1 + Log[1 - x]]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Sum[(j - 1)! Abs[StirlingS1[k, j]], {j, 1, k}] a[n - k]/k!, {k, 1, n}]; Table[n! a[n], {n, 0, 20}]
  • PARI
    x = 'x + O('x^30); Vec(serlaplace(1/(1 + log(1 + log(1 - x))))) \\ Michel Marcus, May 31 2018

Formula

a(n) ~ n! / (exp(2 - exp(-1)) * (1 - exp(exp(-1) - 1))^(n+1)). - Vaclav Kotesovec, May 31 2018
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007840(k). - Seiichi Manyama, May 11 2023

A346944 Expansion of e.g.f. log( 1 + log(1 + x)^2 / 2 ).

Original entry on oeis.org

1, -3, 8, -20, 49, -189, 1791, -21132, 228306, -2274690, 22190772, -230289696, 2756380782, -38757988710, 608149754538, -10057914084048, 171037444641816, -3000345245061048, 55157102668064592, -1077263181846230400, 22411300073192730360, -492846784406541548280
Offset: 2

Views

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

Crossrefs

Cf. A000254, A003713, A081048, A346945, A346946, A346947.
Cf. A346974.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = StirlingS1[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]

Formula

a(n) = Stirling1(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,2) * k * a(k).
a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1) * (2*k)! * Stirling1(n,2*k)/(k * 2^k). - Seiichi Manyama, Jan 23 2025

A346945 Expansion of e.g.f. log( 1 + log(1 + x)^3 / 3! ).

Original entry on oeis.org

1, -6, 35, -235, 1834, -16352, 164044, -1830630, 22513326, -302700926, 4419167532, -69637654996, 1178377833424, -21315571470320, 410529985172400, -8388475139138320, 181270810764205440, -4130796696683135280, 99008773205008777760, -2490134250475836315120
Offset: 3

Views

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

Crossrefs

Cf. A000399, A003713, A081051, A346944, A346946, A346947.
Cf. A346975.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Log[1 + Log[1 + x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS1[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 22}]

Formula

a(n) = Stirling1(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,3) * k * a(k).
a(n) = Sum_{k=1..floor(n/3)} (-1)^(k-1) * (3*k)! * Stirling1(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A346947 Expansion of e.g.f. log( 1 + log(1 + x)^5 / 5! ).

Original entry on oeis.org

1, -15, 175, -1960, 22449, -269451, 3423860, -46238280, 664233856, -10143487354, 164423204582, -2823783679080, 51273355515264, -982236541934430, 19809898439192946, -419752648063849626, 9325875631405818996, -216846992855331506052, 5267598064689049209252
Offset: 5

Views

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

Crossrefs

Cf. A000482, A003713, A346944, A346945, A346946.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS1[n, 5] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 23}]

Formula

a(n) = Stirling1(n,5) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,5) * k * a(k).
a(n) = Sum_{k=1..floor(n/5)} (-1)^(k-1) * (5*k)! * Stirling1(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A373855 a(n) = Sum_{k=1..n} k! * k^(n-1) * |Stirling1(n,k)|.

Original entry on oeis.org

0, 1, 5, 80, 2690, 155074, 13658386, 1706098008, 286888266696, 62485391828448, 17112247116585744, 5755236604915060944, 2331975856351260982848, 1120439648590390138640304, 629855675998212293917375344, 409557081242059531918330384896
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2024

Keywords

Crossrefs

Cf. A003713, A373856.
Cf. A092552, A244585, A373857.
Cf. A320096.

Programs

  • Mathematica
    nmax=15; Range[0,nmax]!CoefficientList[Series[Sum[(-Log[1 - k*x])^k / k,{k,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 19 2024 *)
  • PARI
    a(n) = sum(k=1, n, k!*k^(n-1)*abs(stirling(n, k, 1)));

Formula

E.g.f.: Sum_{k>=1} (-log(1 - k*x))^k / k.

A052809 A simple grammar: number of cycles of cycles.

Original entry on oeis.org

0, 0, 2, 6, 28, 175, 1368, 12838, 140656, 1762794, 24878320, 390495336, 6748280064, 127324033824, 2604355096224, 57404425654080, 1356401049662208, 34202807058719568, 916723959720053760
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A003713, A185393.

Programs

  • Maple
    spec := [S,{B=Cycle(C),C=Cycle(Z),S=Prod(B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[-Log[1+Log[1-x]]*x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)
  • PARI
    a(n) = n*sum(k=1, n-1, (k-1)!*abs(stirling(n-1, k, 1))); \\ Seiichi Manyama, May 20 2022

Formula

E.g.f.: log(-1/(-1+log(-1/(-1+x))))*x.
E.g.f.: -log(1+log(1-x))*x. - Vaclav Kotesovec, Oct 01 2013
a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^(n-1). - Vaclav Kotesovec, Oct 01 2013
a(n) = n * Sum_{k=1..n-1} (k-1)! * |Stirling1(n-1,k)| = n * A003713(n-1). - Seiichi Manyama, May 20 2022

A074708 -log(1-Sum_{n>0} x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.

Original entry on oeis.org

1, 3, 25, 406, 10856, 432536, 24032380, 1776015088, 168482342208, 19958023887552, 2887514448643584, 501074299946343168, 102727197888801668352, 24564844379606631001344, 6776711942017520895558912, 2136315270266212479331055616, 763191034622566084583650197504
Offset: 1

Views

Author

Vladeta Jovovic, Sep 04 2002

Keywords

Links

Crossrefs

Cf. A003713.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[-Log[1 - Sum[x^k/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!^2] (* Vaclav Kotesovec, Jul 17 2021 *)
  • PARI
    seq(n)={Vec(serlaplace(serlaplace(-log(O(x*x^n) + 1 - sum(k=1, n, x^k/k^2)))))} \\ Andrew Howroyd, Jan 27 2020

Formula

a(n) = ((n-1)!)^2 + (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (k-1)!)^2 * (n-k) * a(n-k). - Ilya Gutkovskiy, Jul 16 2021
a(n) ~ d^n * n! * (n-1)!, where d = 1.31312358308891164912353600222812555333413518214112353115268393... - Vaclav Kotesovec, Jul 17 2021

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 27 2020
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