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 1-7 of 7 results.

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

A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.

Original entry on oeis.org

1, 3, 11, 57, 359, 2793, 25871, 273297, 3268199, 44132313, 659178431, 10710083937, 189256343639, 3636935896233, 75228664345391, 1657133255788977, 38770903634692679, 964609458391250553, 25470259163197390751, 709595190213796188417
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 11 2019

Keywords

Links

Crossrefs

Cf. A000005, A000629, A002746, A008277, A028342, A308554, A318249, A330352, A330353, A330354, A330445.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[-Sum[Log[1 - (Exp[x] - 1)^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

Formula

E.g.f.: Sum_{i>=1} Sum_{j>=1} (exp(x) - 1)^(i*j) / (i*j).
E.g.f.: log(Product_{k>=1} 1 / (1 - (exp(x) - 1)^k)^(1/k)).
G.f.: Sum_{k>=1} (k - 1)! * tau(k) * x^k / Product_{j=1..k} (1 - j*x), where tau = A000005.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * tau(k).
a(n) ~ n! * (log(n) + 2*gamma - log(2) - log(log(2))) / (n * (log(2))^n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 14 2019

A330352 Expansion of e.g.f. -Sum_{k>=1} log(1 - log(1 + x)^k) / k.

Original entry on oeis.org

1, 1, 0, 10, -68, 818, -9782, 130730, -1835752, 27408672, -438578616, 7697802264, -150743052528, 3293454634416, -78787556904864, 2014008113598432, -54001416897306240, 1504891127666322048, -43527807706621236480, 1311515508480252542208
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 11 2019

Keywords

Links

Crossrefs

Cf. A000005, A002744, A008275, A028342, A089064, A318249, A330351, A330353, A330354, A330493.

Programs

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

Formula

E.g.f.: Sum_{i>=1} Sum_{j>=1} log(1 + x)^(i*j) / (i*j).
E.g.f.: log(Product_{k>=1} 1 / (1 - log(1 + x)^k)^(1/k)).
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * tau(k), where tau = A000005.

A330494 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 4, 19, 129, 1018, 9912, 111074, 1416398, 20295208, 323437728, 5657339928, 107765338920, 2223272444976, 49399021063584, 1175549092374672, 29822113966614768, 803485297880792064, 22917198585269729664, 689927737384840662144, 21861972842959846530432
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 16 2019

Keywords

Links

Crossrefs

Cf. A330353, A330354, A330493, A330495.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1/(1 - x)]^k/(k (1 - Log[1/(1 - x)]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
  • PARI
    a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*sigma(k)); \\ Michel Marcus, Dec 16 2019

Formula

E.g.f.: Sum_{k>=1} log(1/(1 - x))^k / (k * (1 - log(1/(1 - x))^k)).
a(n) ~ n! * Pi^2 * exp(n) / (6 * (exp(1) - 1)^(n+1)).

A330449 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)^2).

Original entry on oeis.org

1, 6, 36, 282, 2460, 25506, 299796, 3921882, 56977740, 913248786, 15917884356, 299358495882, 6066180049020, 131932872768066, 3057940695635316, 75151035318996282, 1954299203147952300, 53684552455571903346, 1553161560008013680676, 47162101103528811791082
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 15 2019

Keywords

Links

Crossrefs

Cf. A000219, A001157, A008277, A064602, A306046, A318250, A330353, A330450.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]

Formula

E.g.f.: -Sum_{k>=1} k * log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306046.
G.f.: Sum_{k>=1} (k - 1)! * sigma_2(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma_2 = A001157.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000219.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma_2(k).
a(n) ~ n! * zeta(3) * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Dec 15 2019

A330387 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 2, 12, 62, 420, 3782, 40572, 463262, 5708820, 80773622, 1319927532, 23675250062, 447145154820, 8830952572262, 185694817024092, 4246473212654462, 105754322266866420, 2811068529133151702, 78039884046777282252, 2243558766132057764462
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2019

Keywords

Crossrefs

Cf. A000009, A000593, A000629, A008277, A088311, A265024, A305550, A330353, A330388.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) (Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + (Exp[x] - 1)^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Dec 15 2019 *)

Formula

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^(2*k - 1)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A305550.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000009.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * A000593(k).
E.g.f.: Sum_{k>=1} log(1 + (exp(x) - 1)^k). - Vaclav Kotesovec, Dec 15 2019
a(n) ~ n! * Pi^2 / (24 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 15 2019

A330444 a(n) = Sum_{k=1..n} Stirling2(n,k) * (k-1)! * phi(k), where phi = A000010.

Original entry on oeis.org

1, 2, 8, 44, 332, 2852, 28268, 330164, 4371452, 62867492, 980090828, 16792404884, 316446118172, 6484254233732, 142335512881388, 3299266086185204, 80092968046706492, 2040940536907449572, 55097942635383719948, 1586719679112182359124
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 15 2019

Keywords

Links

Crossrefs

Cf. A000010, A330351, A330353.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k] * (k-1)! * EulerPhi[k], {k, 1, n}], {n, 1, 20}]
  • PARI
    a(n) = sum(k=1, n, stirling(n, k, 2)*(k-1)!*eulerphi(k)); \\ Michel Marcus, Dec 15 2019

Formula

a(n) ~ 3 * n! / (Pi^2 * (log(2))^(n+1)).
Showing 1-7 of 7 results.

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