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.

A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 4, 18, 112, 810, 7144, 73458, 850672, 11069370, 161190904, 2575237698, 44571447232, 836188737930, 16970931765064, 368985732635538, 8524290269083792, 208874053200038490, 5428866923032585624, 149250273758730282978, 4318265042184721248352
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 11 2019

Keywords

Links

Crossrefs

Cf. A000041, A000203, A000629, A002745, A008277, A038048, A167137, A308555, A330351, A330352, A330354.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(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)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]

Formula

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A167137.
G.f.: Sum_{k>=1} (k - 1)! * sigma(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma = A000203.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma(k).
a(n) ~ n! * Pi^2 / (12 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 14 2019

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

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.

A330493 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * tau(k), where tau = A000005.

Original entry on oeis.org

1, 3, 12, 70, 492, 4298, 43894, 514666, 6830888, 101473632, 1664125944, 29858266392, 582481147440, 12281821373040, 278257595964576, 6739505703156192, 173785740554811264, 4754455742416944000, 137571331202872821504, 4197696814883284962048
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 16 2019

Keywords

Links

Crossrefs

Cf. A330351, A330352, A330494, A330495.

Programs

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

Formula

E.g.f.: -Sum_{k>=1} log(1 - log(1/(1 - x))^k) / k.
a(n) ~ n! * (log(n) + 2*gamma - log(exp(1) - 1)) / (n * (1 - exp(-1))^n), where gamma is the Euler-Mascheroni constant A001620.

A330445 Expansion of e.g.f.: Sum_{k>=1} log(1 + (exp(x) - 1)^k)/k.

Original entry on oeis.org

0, 1, 1, 5, 19, 89, 691, 7265, 74299, 722489, 8224291, 130439825, 2456898379, 45287950889, 781106871091, 13479917085185, 268959501687259, 6688186010251289, 187628967639969091, 5285049770439071345, 144061583071243096939
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 15 2019

Keywords

Links

Crossrefs

Cf. A330351.

Programs

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

Formula

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

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 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.