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 1-8 of 8 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

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

A346923 Expansion of e.g.f. 1 / (1 - log(1 - x)^4 / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6839, 69804, 784580, 9680000, 130312336, 1901581968, 29895585356, 503657235900, 9051009737834, 172807817059664, 3493189152511608, 74530548004474584, 1673793045085649146, 39467836062718058100, 974939402596817961050, 25177327470510057799550
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346921, A346922, A346924.
Cf. A000454, A346895, A347003, A353119.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - Log[1 - x]^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 4]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^4/4!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/24^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) ~ n! * 2^(-5/4) * 3^(1/4) / (exp(2^(3/4)*3^(1/4)) * (1 - exp(-2^(3/4)*3^(1/4)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/24^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

A353200 Expansion of e.g.f. 1/(1 + log(1 - x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 35947800, 609615600, 12504927600, 281242996320, 6545492073120, 155873050569600, 3849612346944000, 100588974863402880, 2818516832681523840, 84728757269204858880, 2706516690047188416000
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Cf. A052811, A353118, A353119.
Cf. A346924.

Programs

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

Formula

a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42777, 260590, 1809060, 17418401, 229768539, 3402511476, 50013258750, 706670789371, 9659104177101, 130958047050698, 1834295186003784, 27849428308615221, 472297857494304303, 8856291348143365456, 176841068643273207426
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A000670, A330047, A346894, A346895.
Cf. A000481, A327506, A346924.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[1/(1 - (Exp[x] - 1)^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^5/5!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (5*k)!*x^(5*k)/(120^k*prod(j=1, 5*k, 1-j*x)))) \\ Seiichi Manyama, May 09 2022
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/120^k); \\ Seiichi Manyama, May 09 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) ~ n! / (5*(1 + 120^(-1/5)) * log(1 + 120^(1/5))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 09 2022: (Start)
G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(120^k * Product_{j=1..5*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/120^k. (End)

A354394 Expansion of e.g.f. 1/(1 + (exp(x) - 1)^5 / 120).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -15, -140, -1050, -6951, -42273, -232870, -949740, 2401399, 149618469, 2979464124, 47639256210, 683529622229, 9045426379611, 109599657976942, 1148191101672384, 8033814119097459, -50834295574038207, -3977581842278623216, -119536187842156328034
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Cf. A354391, A354392, A354393.
Cf. A346896, A346924, A354398.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^5/120)))
  • 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, j)*stirling(j, 5, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/(-120)^k);

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(-120)^k.

A354135 Expansion of e.g.f. 1/(1 - log(1 + x)^5/120).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, -15, 175, -1960, 22449, -269073, 3403070, -45510630, 643152796, -9586136560, 150319669136, -2473024029840, 42562037379744, -764017130370276, 14260496108114340, -275877454002406236, 5512350021871343616, -113318466860425703184
Offset: 0

Views

Author

Seiichi Manyama, May 18 2022

Keywords

Crossrefs

Cf. A006252, A354134.
Cf. A346920, A346924.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling1(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling1(n,5*k)/120^k.
Showing 1-8 of 8 results.

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