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 1-6 of 6 results.

A305986 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k/k).

Original entry on oeis.org

1, 1, 4, 21, 144, 1205, 11908, 135597, 1745488, 25045821, 396249564, 6850289765, 128438323720, 2595394603269, 56224162108468, 1299717221807229, 31931915643021504, 830816659779428525, 22820190255069409804, 659845945466402034165, 20034230527927369097848, 637252918691725377815349
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 15 2018

Keywords

Comments

Stirling transform of A007841.

Links

Crossrefs

Cf. A007841, A140585, A167137, A305987.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(combinat[multinomial](n, n-i*j, i$j)*
      b(n-i*j, i-1)*(i-1)!^j, j=0..n/i)))
    end:
a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(Exp[x] - 1)^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, #^(1 - k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (exp(x) - 1)^(j*k)/(k*j^k)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A007841(k).
a(n) ~ c * n! * n / log(2)^n, where c = exp(-gamma) / (4*log(2)^2) = 0.29215... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019

A305547 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k!).

Original entry on oeis.org

1, 1, 2, 8, 37, 182, 1039, 7149, 56382, 479220, 4280247, 40406984, 410453366, 4539623168, 54431372233, 695801259947, 9312538336475, 128985882874288, 1842668013046405, 27238267120063415, 419396473955088310, 6769168354222927254, 114837651830425810381, 2042782103293394499566
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 15 2018

Keywords

Comments

Stirling transform of A007837.

Links

Crossrefs

Cf. A007837, A140585, A305987.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
    end:
a:= n-> add(Stirling2(n, k)*b(k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/((j!)^k k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, -# (-#!)^(-k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(exp(x) - 1)^(j*k)/((j!)^k*k)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A007837(k).

A326884 E.g.f.: Product_{k>=1} (1 + k*(exp(x)-1)^k).

Original entry on oeis.org

1, 1, 5, 43, 377, 4291, 58745, 914803, 15641897, 298104451, 6337624985, 147137420563, 3674045105417, 98093008751011, 2793940490888825, 84812168406518323, 2737609202984488937, 93486719521251467971, 3358396276982001106265, 126434158646122122080083
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 31 2019

Keywords

Links

Crossrefs

Cf. A022629, A305550, A305987, A326885.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1+k*(Exp[x]-1)^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) = Sum_{k=0..n} A022629(k)*Stirling2(n,k)*k!.

A326887 E.g.f.: Product_{k>=1} (1 + (exp(x)-1)^k/k) / (1 - (exp(x)-1)^k/k).

Original entry on oeis.org

1, 2, 8, 48, 364, 3320, 35464, 433692, 5962548, 90931152, 1522657264, 27765229844, 547487475484, 11604952395816, 263091290017560, 6351255101776812, 162643987129698628, 4403250400372110656, 125649232950852714496, 3769013390615951560068, 118555772298034094231724
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 31 2019

Keywords

Links

Crossrefs

Cf. A305199, A305986, A305987.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1+(Exp[x]-1)^k/k)/(1-(Exp[x]-1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) = Sum_{k=0..n} A305199(k)*Stirling2(n,k).
a(n) ~ n * (n+1)! / (16 * exp(2*gamma) * log(2)^(n+3)), where gamma is the Euler-Mascheroni constant A001620.

A345750 E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k).

Original entry on oeis.org

1, 1, 2, 9, 49, 310, 2521, 25557, 290550, 3555041, 48104901, 741103946, 12825399313, 240202011881, 4747281446090, 98808864563065, 2194031697420057, 52582450760730398, 1357237338948268649
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2021

Keywords

Comments

Stirling transform of A168243.

Links

Crossrefs

Cf. A048272, A048993, A168243, A305550, A305987, A345749, A345751.

Programs

  • Mathematica
    max = 18; Range[0, max]! * CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k)^(1/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+(exp(x)-1)^k)^(1/k))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, -sumdiv(k, d, (-1)^d)*(exp(x)-1)^k/k))))

Formula

E.g.f.: exp( Sum_{k>=1} A048272(k) * (exp(x) - 1)^k / k ).
a(n) = Sum_{k=0..n} Stirling2(n,k) * A168243(k).

A345752 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k / k).

Original entry on oeis.org

1, -1, -2, -3, 0, 55, 572, 4865, 40912, 351675, 2978196, 23418373, 148849544, 84185855, -27459134420, -881482705719, -21652972750464, -487503384038525, -10785437160748156, -242968902040697011, -5627949704687484872, -133358411031825299385
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2021

Keywords

Comments

Stirling transform of A292359.

Links

Crossrefs

Cf. A048993, A292359, A305986, A305987, A345751.

Programs

  • Mathematica
    max = 21; Range[0, max]! * CoefficientList[Series[Product[1 - (Exp[x] - 1)^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k/k)))

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A292359(k).
Showing 1-6 of 6 results.

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