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 1-3 of 3 results.

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

A306037 Expansion of e.g.f. 1/(1 + log(1 - log(1 + x))).

Original entry on oeis.org

1, 1, 2, 7, 31, 178, 1200, 9588, 86592, 887086, 10035164, 125472246, 1705102394, 25175822644, 399387494956, 6801042408728, 123348694663480, 2379855020533664, 48569042602254128, 1047134236970183664, 23748242269316806752, 565834452464428045872, 14117321495269290091440
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 17 2018

Keywords

Examples

			1/(1 + log(1 - log(1 + x))) = 1 + x + 2*x^2/2! + 7*x^3/3! + 31*x^4/4! + 178*x^5/5! + 1200*x^6/6! + ...

Links

Crossrefs

Cf. A007840, A089064, A305323, A305988.

Programs

  • Maple
    a:=series(1/(1+log(1-log(1+x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + Log[1 - Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[StirlingS1[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 22}]
a[0] = 1; a[n_] := a[n] = Sum[Sum[(j - 1)! StirlingS1[k, j], {j, 1, k}] a[n - k]/k!, {k, 1, n}]; Table[n! a[n], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A007840(k).
a(n) ~ n! * exp(-exp(-1)) / (exp(1 - exp(-1)) - 1)^(n+1). - Vaclav Kotesovec, Jul 01 2018

A361494 Expansion of e.g.f. 1/(1 - log(2 - exp(x))).

Original entry on oeis.org

1, -1, 0, 0, -2, -10, -62, -518, -5042, -55914, -700982, -9801022, -151141850, -2548546130, -46648614014, -921144036486, -19518279101570, -441740723440186, -10635049333176902, -271391755745104334, -7317268150934309162, -207850529950047641250
Offset: 0

Views

Author

Seiichi Manyama, May 11 2023

Keywords

Crossrefs

Cf. A217033, A360446.
Cf. A006252, A305988.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-Log[2-Exp[x]]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 04 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(2-exp(x)))))

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * A006252(k).
Showing 1-3 of 3 results.

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