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 1-4 of 4 results.

A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).

Original entry on oeis.org

1, 0, 0, 1, 6, 25, 110, 721, 6286, 57625, 541470, 5558641, 64351166, 819480025, 11140978030, 160711583761, 2472834185646, 40597082635225, 706816137889790, 12974021811748081, 250395124862965726, 5074637684604691225, 107798916619788396750
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

Links

Crossrefs

Cf. A000670, A330047, A346895, A346920.
Cf. A000392, A327504, A346922, A353774.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - (Exp[x] - 1)^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[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-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 06 2021
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(6^k*prod(j=1, 3*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022
  • 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, 3, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/6^k); \\ Seiichi Manyama, May 07 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
a(n) ~ n! / (3*(1 + 6^(-1/3)) * log(1 + 6^(1/3))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(6^k * Product_{j=1..3*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/6^k. (End)

A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 65, 350, 1771, 10290, 86605, 977350, 11778041, 138208070, 1590920695, 18895490250, 245692484311, 3587464083850, 57397496312585, 966066470023550, 16713560617838581, 297182550111615630, 5500448659383161275, 107267326981597659250
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

Links

Crossrefs

Cf. A000670, A330047, A346894, A346920.
Cf. A000453, A327505, A346923, A353775.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - (Exp[x] - 1)^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[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-(exp(x)-1)^4/4!))) \\ Michel Marcus, Aug 06 2021
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(24^k*prod(j=1, 4*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022
  • 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, 4, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/24^k); \\ Seiichi Manyama, May 07 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
a(n) ~ n! / (4*(1 + 2^(-3/4)*3^(-1/4)) * log(1 + 2^(3/4)*3^(1/4))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(24^k * Product_{j=1..4*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/24^k. (End)

A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

Original entry on oeis.org

1, 15, 140, 1050, 6951, 42651, 253660, 1594230, 12463451, 134921787, 1806513072, 25539589530, 355175465191, 4797717669123, 63797550625676, 860468790181686, 12275324511112971, 192498455326842819, 3353266112959628272, 63379650000684213834
Offset: 5

Views

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

Crossrefs

Cf. A000481, A000629, A003704, A327506, A346390, A346920, A346954.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

Formula

a(n) = Stirling2(n,5) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ (n-1)! / (log(120^(1/5) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

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 1-4 of 4 results.

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