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 1-9 of 9 results.

A346922 Expansion of e.g.f. 1 / (1 + log(1 - x)^3 / 3!).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 245, 2044, 19572, 210524, 2513760, 33012276, 472963876, 7340889192, 122703087416, 2197496734224, 41979155247520, 852063971170960, 18312093589455440, 415420659953439840, 9920128280950954080, 248735658391768241280, 6533773435848445617600
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346921, A346923, A346924.
Cf. A000399, A346894, A347002, A353118.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + Log[1 - x]^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 3]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^3/3!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/6^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
a(n) ~ n! * 6^(1/3) / (3 * exp(6^(1/3)) * (1 - exp(-6^(1/3)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/6^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

A347003 Expansion of e.g.f. exp( log(1 - x)^4 / 4! ).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6804, 68544, 754130, 9044750, 117773656, 1656897528, 25061576176, 405667844400, 6997383182854, 128126051451184, 2481884332498848, 50702417505257904, 1089371806098805286, 24555007848629510700, 579348221233739760550, 14278529041496660104450
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A000454, A327505, A346923, A347001, A347002, A347004.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Log[1 - x]^4/4!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 4]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/(24^k*k!)); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,4)| * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/(24^k * k!). - Seiichi Manyama, May 06 2022

A347004 Expansion of e.g.f. exp( -log(1 - x)^5 / 5! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269451, 3423860, 46238280, 664233856, 10143487354, 164423078456, 2823768543960, 51272283444264, 982177492263750, 19807082824819374, 419629806223448346, 9320808413229618816, 216645165604679499072, 5259724543984442886486
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A000482, A327506, A346924, A347001, A347002, A347003.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[-Log[1 - x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/(120^k*k!)); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = 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)|/(120^k * k!). - Seiichi Manyama, May 06 2022

A353344 Expansion of e.g.f. exp(-log(1 - x)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 1710, 17304, 194712, 2402184, 32536080, 481094856, 7703580456, 132658888752, 2443228469136, 47904722262144, 995970495769920, 21879712141853760, 506301721998264000, 12306713585213260800, 313441368701926135680, 8345931596469584686080
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=3 of A357882.
Cf. A000142, A353358, A353404.
Cf. A347002, A353118, A353664.

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)*abs(stirling(j, 3, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*abs(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!.

A353892 Expansion of e.g.f. exp( -(x * log(1-x))^3 / 36 ).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 20, 210, 1960, 18900, 194880, 2166780, 26172080, 342599400, 4835694864, 73208215080, 1183011385920, 20318534134080, 369549843420384, 7094851788127680, 143377043010268800, 3042204544957939200, 67621161484919380800, 1571319471977711258880
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A066166, A353891, A353893.
Cf. A347002, A353881, A353895.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(x*log(1-x))^3/36)))
  • PARI
    a(n) = n!*sum(k=0, n\6, (3*k)!*abs(stirling(n-3*k, 3*k, 1))/(36^k*k!*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/6)} (3*k)! * |Stirling1(n-3*k,3*k)|/(36^k * k! * (n-3*k)!).

A357037 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(log(1 - x * A(x))^2 / 6).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 295, 3304, 42112, 599724, 9657330, 174222576, 3464835726, 75208002792, 1771121398956, 44998593873024, 1226723273550720, 35714547582173280, 1106012915718532920, 36304411160854523520, 1259105580819317636280, 46007354360033491345920
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2022

Keywords

Crossrefs

Cf. A001761, A357036.
Cf. A347002, A357029, A357032.

Programs

  • Mathematica
    m = 22; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x*A[x])^(Log[1 - x*A[x]]^2/6) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*(n+1)^(k-1)*abs(stirling(n, 3*k, 1))/(6^k*k!));

Formula

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

A354136 Expansion of e.g.f. exp(log(1 + x)^3/6).

Original entry on oeis.org

1, 0, 0, 1, -6, 35, -215, 1414, -9912, 73044, -552570, 4102626, -26654826, 79506492, 2154425364, -73527421176, 1708053626880, -35961691589640, 736338276883080, -15067241745943680, 312009998091705720, -6579362641255341120, 141704946709227843480
Offset: 0

Views

Author

Seiichi Manyama, May 18 2022

Keywords

Crossrefs

Cf. A354137.
Cf. A327504, A347002, A354134.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^3/6)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=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)/(6^k*k!));

Formula

a(0) = 1; a(n) = 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)/(6^k * k!).

A357883 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)|/(k!^j * j!).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 6, 0, 1, 0, 0, 3, 24, 0, 1, 0, 0, 1, 14, 120, 0, 1, 0, 0, 0, 6, 80, 720, 0, 1, 0, 0, 0, 1, 35, 544, 5040, 0, 1, 0, 0, 0, 0, 10, 235, 4284, 40320, 0, 1, 0, 0, 0, 0, 1, 85, 1834, 38310, 362880, 0, 1, 0, 0, 0, 0, 0, 15, 735, 16352, 383256, 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,  1,  0,  0, 0, ...
  0,   6,  3,  1,  0, 0, ...
  0,  24, 14,  6,  1, 0, ...
  0, 120, 80, 35, 10, 1, ...

Crossrefs

Columns k=0-5 give: A000007, A000142, A347001, A347002, A347003, A347004.
Cf. A324162, A357119, A357881, A357882.

Programs

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

Formula

For k > 0, e.g.f. of column k: exp((-log(1-x))^k / k!).
T(0,k) = 1; T(n,k) = Sum_{j=1..n} binomial(n-1,j-1) * |Stirling1(j,k)| * T(n-j,k).
Showing 1-9 of 9 results.

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