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.

A320349 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k).

Original entry on oeis.org

1, 1, 5, 32, 278, 2894, 35986, 514128, 8306448, 149558688, 2968216944, 64314676128, 1510065781968, 38178537908016, 1033794746169168, 29840453678758272, 914461132860063360, 29645845798652997120, 1013511411165693991680, 36436289007997132646400, 1373976152501162688288000
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 11 2018

Keywords

Links

Crossrefs

Cf. A000041, A000203, A048994, A053529, A167137, A306042, A320350.

Programs

  • Maple
    seq(n!*coeff(series(mul(1/(1-log(1/(1-x))^k),k=1..100),x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] PartitionsP[k] k!, {k, 0, n}], {n, 0, 20}]

Formula

E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1/(1 - x))^k/k).
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000041(k)*k!.
From Vaclav Kotesovec, Oct 13 2018: (Start)
a(n) ~ n! * exp(n + Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) / (4 * sqrt(3) * n * (exp(1) - 1)^n).
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) * n^(n - 1/2) / (2^(3/2) * sqrt(3) * (exp(1) - 1)^n).
(End)

A330354 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)).

Original entry on oeis.org

1, 2, 1, 21, -122, 1752, -21730, 309166, -4521032, 70344768, -1173530712, 21642745704, -448130571696, 10352684535840, -260101132095888, 6921279885508848, -191813249398678272, 5502934340821289088, -163695952380982280832, 5078687529186002247552
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 11 2019

Keywords

Links

Crossrefs

Cf. A000041, A000203, A002743, A008275, A038048, A089064, A306042, A330351, A330352, A330353, A330494.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]

Formula

E.g.f.: -Sum_{k>=1} log(1 - log(1 + x)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306042.
exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma(k), where sigma = A000203.
Conjecture: a(n) ~ n! * (-1)^n * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Dec 16 2019

A298905 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).

Original entry on oeis.org

1, 1, 1, 8, -8, 224, -712, 9120, -53496, 980088, -14394648, 264140832, -4113747024, 59028225840, -545558201424, -4191307074432, 450100910950272, -17302659472138752, 530508727766191104, -14790496500550616832, 408513443917280375808, -12274212131738107257600
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 18 2018

Keywords

Links

Crossrefs

Cf. A000009, A088311, A305550, A306023, A306042, A320350.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(Stirling1(n, j)*b(j)*j!, j=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 18 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]

Formula

E.g.f.: exp(Sum_{k>=1} (-1)^(k+1)*log(1 + x)^k/(k*(1 - log(1 + x)^k))).
a(n) = Sum_{k=0..n} Stirling1(n,k)*A000009(k)*k!.

A308878 Expansion of e.g.f. (1 - log(1 + x))/(1 - 2*log(1 + x)).

Original entry on oeis.org

1, 1, 3, 14, 86, 664, 6136, 66240, 816672, 11331552, 174662304, 2961774144, 54785368128, 1097882522112, 23693117756928, 547844658441216, 13511950038494208, 354086653712228352, 9824794572366544896, 287752569360558907392, 8871374335098501292032
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

Comments

Inverse Stirling transform of A002866.

Crossrefs

Cf. A002866, A008275, A011782, A050351, A088501, A306042, A308877.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - Log[1 + x])/(1 - 2 Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] 2^(k - 1) k!, {k, 1, n}], {n, 1, 20}]]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} Stirling1(n,k) * 2^(k-1) * k!.
a(n) ~ n! * exp(1/2) / (4 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Jun 29 2019

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).

A307523 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k)/(1 - log(1 + x)^k).

Original entry on oeis.org

1, 2, 6, 28, 124, 848, 5312, 40080, 367632, 3132096, 27731328, 474979008, 1130161728, 90279554688, 268809015168, 3005011325952, 473192066191104, -7913323872693504, 186235895195313408, 1357401816746159616, -181477915903332002304, 9552839425392612096000
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2019

Keywords

Comments

Exponential convolution of A298905 and A306042.

Crossrefs

Cf. A015128, A298905, A306042, A306045, A307524.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k)/(1 - Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 21}]

Formula

E.g.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*log(1 + x)^k/k).
E.g.f.: 1/theta_4(log(1 + x)).
a(n) = Sum_{k=0..n} Stirling1(n,k)*A015128(k)*k!.
Showing 1-6 of 6 results.

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