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.

A354240 Expansion of e.g.f. 1/sqrt(1 - 4 * log(1+x)).

Original entry on oeis.org

1, 2, 10, 88, 1080, 17088, 330528, 7558752, 199487136, 5967529152, 199533657792, 7374470138880, 298520508249600, 13135454575464960, 624240306760343040, 31864146725023718400, 1738698154646011499520, 100996114388088994007040
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A009199, A320343, A354241, A354242, A354243.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/Sqrt[1-4Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 04 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-4*log(1+x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*log(1+x)^k)))
  • PARI
    a(n) = sum(k=0, n, (2*k)!*stirling(n, k, 1)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * log(1+x)^k.
a(n) = Sum_{k=0..n} (2*k)! * Stirling1(n,k)/k!.
a(n) ~ n^n / (sqrt(2) * (exp(1/4)-1)^(n + 1/2) * exp(n - 1/8)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A357882 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 6, 0, 1, 0, 0, 6, 24, 0, 1, 0, 0, 6, 34, 120, 0, 1, 0, 0, 0, 36, 220, 720, 0, 1, 0, 0, 0, 24, 210, 1688, 5040, 0, 1, 0, 0, 0, 0, 240, 1710, 14868, 40320, 0, 1, 0, 0, 0, 0, 120, 2040, 17304, 147684, 362880, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 194712, 1631376, 3628800, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   0,   0,   0,   0, ...
  0,   2,   2,   0,   0,   0, ...
  0,   6,   6,   6,   0,   0, ...
  0,  24,  34,  36,  24,   0, ...
  0, 120, 220, 210, 240, 120, ...

Crossrefs

Columns k=0-5 give: A000007, A000142, (-1)^n * A009199(n), A353344, A353358, A353404.
Cf. A357119, A357869, A357881.

Programs

  • PARI
    T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1))/j!);
  • PARI
    T(n, k) = if(k==0, 0^n, n!*polcoef(exp((-log(1-x+x*O(x^n)))^k), n));

Formula

For k > 0, e.g.f. of column k: exp((-log(1-x))^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * |Stirling1(j,k)| * T(n-j,k).

A308346 Expansion of e.g.f. 1/(1 - x)^log(1 - x).

Original entry on oeis.org

1, 0, -2, -6, -10, 20, 352, 2772, 18132, 104400, 469608, 238920, -35811048, -730972944, -11436661728, -164609993520, -2294024595312, -31488879303552, -426338226719904, -5626751283423072, -70000948158061728, -745703905072996800, -4142683990211677440, 110386551348875714880
Offset: 0

Views

Author

Ilya Gutkovskiy, May 21 2019

Keywords

Links

Crossrefs

Cf. A000254, A008405, A009199, A052517, A067994, A086660, A087761.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^Log(1/(1-x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 21 2019
  • Maple
    E:= 1/(1-x)^log(1-x):
S:= series(E,x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, May 22 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - x)^Log[1 - x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HermiteH[k, 0], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = -2 Sum[(k - 1)! HarmonicNumber[k - 1] Binomial[n - 1, k - 1] a[n - k], {k, 2, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
  • PARI
    a(n) = sum(k=0, n, abs(stirling(n, k, 1))*polhermite(k, 0)); \\ Michel Marcus, May 21 2019
  • Sage
    m = 30; T = taylor((1-x)^log(1/(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 21 2019

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A067994(k).

A354231 Expansion of e.g.f. exp(log(1 + x)^3).

Original entry on oeis.org

1, 0, 0, 6, -36, 210, -990, 2184, 37128, -863736, 13020480, -168384744, 1940801544, -18825129648, 107706637584, 1386022834944, -73429347222720, 2034345021802560, -46869707752067520, 976421492688165120, -18675350766042871680, 319467427583225518080
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A009199, A354232.
Cf. A353344, A354136, A354229.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^2))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/k!);

Formula

E.g.f.: (1 + x)^(log(1 + x)^2).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/k!.

A354232 Expansion of e.g.f. exp(log(1 + x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, -1800, 21000, -235200, 2693880, -30504600, 310239600, -2026767600, -22324267680, 1480359360480, -48314853350400, 1332965821824000, -34178451017685120, 837433109548661760, -19671723873906894720, 436228097513559408000
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A009199, A354231.
Cf. A354137, A354230.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^5)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^4))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/k!);

Formula

E.g.f.: (1 + x)^(log(1 + x)^4).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling1(n,5*k)/k!.

A086934 E.g.f.: (sqrt(Pi)/2)*erf(log(1-x)).

Original entry on oeis.org

0, -1, -1, 0, 6, 34, 150, 548, 1064, -9620, -190980, -2379408, -26412408, -281137848, -2923819080, -29496805392, -277842087360, -2151448802448, -5085750338064, 334794720634752, 11987850975483360
Offset: 0

Views

Author

Vladeta Jovovic, Sep 21 2003

Keywords

Links

Crossrefs

Cf. A009199.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Sqrt[Pi]/2 Erf[Log[1-x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 04 2015 *)

Formula

Sum_{k=0..floor((n-1)/2)} (-1)^(n-k)*Stirling1(n, 2*k+1)*(2*k)!/(k)!.

Extensions

Definition clarified by Harvey P. Dale, Mar 04 2015

A308535 Expansion of e.g.f. 1/(1 - x)^log(1 + x) (even powers only).

Original entry on oeis.org

1, 2, 22, 608, 31764, 2695992, 338441112, 58961602464, 13614906576528, 4024831155397536, 1482492491866434912, 665729215100873644800, 358022910151079384324928, 227174478580352888344068480, 167941710127005880795828894080, 143087068385495604780364250426880
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2019

Keywords

Crossrefs

Cf. A008405, A009199, A087761, A308346.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[1/(1 - x)^Log[1 + x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] 1/(1 - x)^log(1 + x).
Showing 1-7 of 7 results.

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