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 1-4 of 4 results.

A346945 Expansion of e.g.f. log( 1 + log(1 + x)^3 / 3! ).

Original entry on oeis.org

1, -6, 35, -235, 1834, -16352, 164044, -1830630, 22513326, -302700926, 4419167532, -69637654996, 1178377833424, -21315571470320, 410529985172400, -8388475139138320, 181270810764205440, -4130796696683135280, 99008773205008777760, -2490134250475836315120
Offset: 3

Views

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

Crossrefs

Cf. A000399, A003713, A081051, A346944, A346946, A346947.
Cf. A346975.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Log[1 + Log[1 + x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS1[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 22}]

Formula

a(n) = Stirling1(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,3) * k * a(k).
a(n) = Sum_{k=1..floor(n/3)} (-1)^(k-1) * (3*k)! * Stirling1(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A346974 Expansion of e.g.f. log( 1 + (exp(x) - 1)^2 / 2 ).

Original entry on oeis.org

1, 3, 4, -15, -134, -357, 2374, 33645, 133186, -1288617, -24887906, -130115895, 1666879306, 40612637523, 262868197414, -4221449488635, -123802488449774, -952293015617937, 18497401668708334, 632675912865355425, 5622243546094977946, -128799294291220310997
Offset: 2

Views

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

Crossrefs

Cf. A000225, A003704, A060311, A346975, A346976, A346977.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = StirlingS2[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]

Formula

a(n) = Stirling2(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,2) * k * a(k).
a(n) ~ -n! * 2^(n+1) * cos(n*arctan(2*arctan(sqrt(2))/log(3))) / (n * (4*arctan(sqrt(2))^2 + log(3)^2)^(n/2)). - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1) * (2*k)! * Stirling2(n,2*k)/(k * 2^k). - Seiichi Manyama, Jan 23 2025

A346976 Expansion of e.g.f. log( 1 + (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1666, 6510, 7855, -270050, -4942894, -63052990, -682650605, -6309889950, -42960995804, 348211510, 7739540496935, 202902567668150, 3863986259609686, 61527382177040010, 807717870749781475, 7066953051021894250, -33781117662453993424
Offset: 4

Views

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

Crossrefs

Cf. A000453, A327505, A346954, A346974, A346975, A346977.
Cf. A346946, A346954.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

Formula

a(n) = Stirling2(n,4) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ -(n-1)! * 2^(n+1) * cos(n*arctan((2*arctan(1/(1 + 1/6^(1/4)))) / log(1 + 2*6^(1/4) + 2*6^(1/2)))) / (4*arctan(1/(1 + 1/6^(1/4)))^2 + log(1 + 2*6^(1/4) + 2*6^(1/2))^2)^(n/2). - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (-1)^(k-1) * (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A346977 Expansion of e.g.f. log( 1 + (exp(x) - 1)^5 / 5! ).

Original entry on oeis.org

1, 15, 140, 1050, 6951, 42399, 239800, 1164570, 2553551, -54771717, -1384474728, -23286667950, -339924740609, -4554547609233, -56481301888144, -630768487283886, -5665064764515849, -18095553874845909, 924820173031946752, 35413415495503624986
Offset: 5

Views

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

Crossrefs

Cf. A000481, A327506, A346955, A346974, A346975, A346976.
Cf. A346947, A346955.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

Formula

a(n) = Stirling2(n,5) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ -(n-1)! * 2^(1+n) * 5^n * cos(n*arctan((2*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))) / log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))) / (100*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))^2 + (5*log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))^2)^(n/2). - Vaclav Kotesovec, Aug 10 2021
a(n) = Sum_{k=1..floor(n/5)} (-1)^(k-1) * (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025
Showing 1-4 of 4 results.

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