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.

Previous Showing 11-17 of 17 results.

A326885 E.g.f.: Product_{k>=1} 1/(1 - k*(exp(x)-1)^k).

Original entry on oeis.org

1, 1, 7, 55, 595, 7351, 110587, 1884415, 36154195, 771983911, 18141124267, 463345240975, 12792709110595, 379854657215671, 12057296962232347, 407072488594360735, 14565548824196479795, 550582832110097346631, 21917855760706255154827, 916261422041320023467695
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 31 2019

Keywords

Links

Crossrefs

Cf. A006906, A167137, A305986, A326884.

Programs

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

Formula

a(n) = Sum_{k=0..n} A006906(k)*Stirling2(n,k)*k!.
a(n) ~ c * n! / ((3^(2/3) - 2) * (3^(2/3) - 1) * log(1 + 3^(-1/3))^(n+1)), where c = Product_{k>=4} 1/(1 - k/3^(k/3)) = 3468.14377687388560106742710672518465524...

A335812 E.g.f.: Product_{k>=1} 1 / (1 - (1 - exp(x))^k).

Original entry on oeis.org

1, -1, 3, -7, 39, -31, 1623, 9953, 182199, 2116289, 32269143, 505278113, 9743069559, 214428606209, 5156280298263, 127321200213473, 3176128419544119, 80737907621585729, 2147513299611040983, 61423058495936864033, 1912348969322283717879, 64216042408215934910849
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 25 2020

Keywords

Comments

Inverse binomial transform of A327601.

Crossrefs

Cf. A000041, A167137, A327601, A335813.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[ Product[1/(1 - (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] k! PartitionsP[k], {k, 0, n}], {n, 0, 21}]
  • PARI
    a(n) = sum(k=0, n, (-1)^k * stirling(n,k,2) * k! * numbpart(k)); \\ Michel Marcus, Jun 25 2020

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A000041(k).

A316145 a(n) = Sum_{k=0..n} Stirling2(n,k) * A000041(k) * k^k.

Original entry on oeis.org

1, 9, 106, 1823, 36821, 932080, 26666067, 876727561, 32137538059, 1305168046976, 57774609056649, 2783202675369037, 144453227105110782, 8035192765567735275, 476686201707606976317, 30053582893540865299197, 2005019178999976881804130, 141111387620531900621281975
Offset: 1

Views

Author

Vaclav Kotesovec, Jun 25 2018

Keywords

Links

Crossrefs

Cf. A167137, A282190, A306022, A316146.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] * PartitionsP[k] * k^k, {k, 1, n}], {n, 1, 20}]

Formula

Limit_{n -> infinity} (a(n)/n!)^(1/n) = 1/(log(1+ exp(1)) - 1) = 3.1922192845297391106277924019427161296056687330974482534324... - Vaclav Kotesovec, Nov 21 2021
log(a(n) / A316146(n)) ~ (sqrt(2) - 1) * Pi * sqrt(n) / sqrt(3*(1 + exp(1)) * log(1 + exp(-1))). - Vaclav Kotesovec, Nov 22 2021

A327601 Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - (1 - exp(x))^k).

Original entry on oeis.org

1, 0, 2, 0, 26, 120, 1922, 21840, 307946, 4251240, 63165842, 1010729280, 18501318266, 391496665560, 9265945721762, 232411950454320, 5972325812958986, 156131611764907080, 4208451299935189682, 119669466221148348960, 3658459009408581118106
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 18 2019

Keywords

Comments

Stirling-Bernoulli transform of partition numbers (A000041).

Crossrefs

Cf. A000041, A167137, A306022, A306042.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[x] Product[1/(1 - (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n + 1, k + 1] k! PartitionsP[k], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, (-1)^k*stirling(n+1, k+1, 2)*k!*numbpart(k)); \\ Michel Marcus, Sep 19 2019

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n+1,k+1) * k! * A000041(k).

A336100 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k).

Original entry on oeis.org

1, -1, -3, -7, -15, 89, 1737, 21713, 266865, 3162089, 34737177, 352100033, 2848598145, -7655375911, -1359369828183, -50221626404047, -1460912626424175, -39804558811289911, -1080962878982246343, -29431779044695154527, -788320672341728128095, -20386762121171790275911
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2020

Keywords

Crossrefs

Cf. A010815, A167137, A335812, A335813, A336097.

Programs

  • Mathematica
    m = 21; Range[0, m]! * CoefficientList[Series[Product[1 - (Exp[x] - 1)^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Jul 08 2020 *)
A010815[k_] := (m = (1 + Sqrt[1 + 24*k])/6; If[IntegerQ[m], (-1)^m, 0] + If[IntegerQ[m - 1/3], (-1)^(m - 1/3), 0]); Table[Sum[StirlingS2[n, k] * k! * A010815[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 09 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k)))
  • PARI
    f(n) = if( issquare( 24*n + 1, &n), kronecker( 12, n)); \\ A010815
a(n) = sum(k=0, n, stirling(n,k,2) * k! * f(k)); \\ Michel Marcus, Jul 09 2020

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * A010815(k).

A336097 E.g.f.: Product_{k>=1} (1 - (1 - exp(x))^k).

Original entry on oeis.org

1, 1, -1, -5, -13, -149, -1861, -21965, -267373, -3163109, -34739221, -352104125, -3806609533, -67068890069, -1866226978981, -51776974365485, -1180415240484493, -19613026052409029, -122604194898649141, 6950364605049945955, 394565422299921179747, 13840685990526765512011
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2020

Keywords

Crossrefs

Cf. A010815, A167137, A335812, A335813, A336100.

Programs

  • Mathematica
    m = 21; Range[0, m]! * CoefficientList[Series[Product[1 - (1 - Exp[x])^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Jul 08 2020 *)
A010815[k_] := (m = (1 + Sqrt[1 + 24*k])/6; If[IntegerQ[m], (-1)^m, 0] + If[IntegerQ[m - 1/3], (-1)^(m - 1/3), 0]); Table[Sum[(-1)^k * StirlingS2[n, k] * k! * A010815[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 09 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1-(1-exp(x))^k)))

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A010815(k).

A345749 E.g.f.: Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^(1/k).

Original entry on oeis.org

1, 1, 4, 21, 147, 1250, 12633, 147497, 1947676, 28699373, 466994003, 8309274754, 160368858609, 3336869582657, 74468098634660, 1773827462044421, 44905503103938915, 1203843692164105458, 34070243272290551113, 1015056385225183643721
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2021

Keywords

Comments

Stirling transform of A028342.

Links

Crossrefs

Cf. A000005, A028342, A048993, A167137, A294363, A305986, A345750, A345751.

Programs

  • Mathematica
    max = 19; Range[0, max]! * CoefficientList[Series[Product[1/(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(1/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,numdiv(k)*(exp(x)-1)^k/k))))

Formula

E.g.f.: exp( Sum_{k>=1} d(k) * (exp(x) - 1)^k / k ), where d(n) is the number of divisors of n.
a(n) = Sum_{k=0..n} Stirling2(n,k) * A028342(k).
Previous Showing 11-17 of 17 results.

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