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 1-10 of 11 results. Next

A343707 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 3, 15, 113, 1145, 14539, 221663, 3943281, 80173345, 1833831619, 46606646175, 1302954958689, 39737420405753, 1312901360002283, 46714233470065999, 1780859204826798401, 72416689888874547969, 3128792006916853876291, 143132514626658326870767, 6911638338982428907738641
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Links

Crossrefs

Cf. A088500, A201339, A291979, A343709, A343710.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 2 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x]/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+2*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / (1 + 2 * log(1 - x)).
a(n) = Sum_{k=0..n} binomial(n,k) * A088500(k).

A330149 Expansion of e.g.f. exp(-x) / (1 + log(1 - x)).

Original entry on oeis.org

1, 0, 2, 7, 47, 368, 3494, 38673, 489341, 6966344, 110199090, 1917589771, 36402276107, 748629861016, 16580304397942, 393443385034069, 9958671117295737, 267824225078212336, 7626444798009902530, 229232204568273395919, 7252798333599466521575, 240948882537990850397536
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2019

Keywords

Comments

Inverse binomial transform of A007840.

Crossrefs

Cf. A052841, A330150.
Cf. A007840, A291979.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[-x]/(1 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A007840(k).
a(n) ~ n! * exp(n + exp(-1) - 1) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Dec 15 2019
a(n) = (-1)^n + Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Dec 19 2023

A343709 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 4, 28, 295, 4159, 73348, 1552468, 38336569, 1081926157, 34350646636, 1211796777748, 47023762576987, 1990643657768683, 91291802205304972, 4508735102829489580, 238583762726054522989, 13466532093135977880025, 807606110028529741369396, 51282242176105846536128236
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Links

Crossrefs

Cf. A201354, A291979, A343707, A343710.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[x]/(1 + 3 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+3*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / (1 + 3 * log(1 - x)).

A343710 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 5, 45, 609, 11009, 248837, 6749629, 213596401, 7725031521, 314310704101, 14209394894765, 706617979262049, 38333841625642785, 2252901018519028901, 142589176837851349757, 9669282207517755852721, 699408060608904410296897, 53752166013267632536864581, 4374061543586452325644329133
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Links

Crossrefs

Cf. A201365, A291979, A343707, A343709.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 4 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[x]/(1 + 4 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+4*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / (1 + 4 * log(1 - x)).

A368283 Expansion of e.g.f. exp(2*x) / (1 + log(1 - x)).

Original entry on oeis.org

1, 3, 11, 52, 320, 2486, 23402, 258252, 3263528, 46433648, 734322672, 12776283136, 242519067056, 4987324250416, 110454579648688, 2621008072506592, 66341399843669760, 1784150447268259456, 50804574646886197888, 1527058892582680257024
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A007840, A291979, A330149, A368284.
Cf. A368285.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=2^i+sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 2^n + Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ n! * exp(n + 2 - 2*exp(-1)) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Dec 29 2023

A368445 Expansion of e.g.f. exp(x) / (1 + log(1 - 3*x)).

Original entry on oeis.org

1, 4, 34, 469, 8815, 208348, 5922118, 196568419, 7459854973, 318560689324, 15116763184978, 789119869380577, 44939583072146251, 2772582488089509028, 184216538154508055062, 13114092114632287359919, 995813104288130697683065, 80342826520464644566291828
Offset: 0

Views

Author

Seiichi Manyama, Dec 24 2023

Keywords

Links

Crossrefs

Cf. A291979, A368444.
Cf. A343709.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[x]/(1+Log[1-3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 07 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=1, i, 3^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 1 + Sum_{k=1..n} 3^k * (k-1)! * binomial(n,k) * a(n-k).

A368450 Expansion of e.g.f. exp(x) / (1 + log(1 - 3*x)/3).

Original entry on oeis.org

1, 2, 8, 61, 695, 10310, 187024, 4002131, 98593949, 2746565218, 85333213856, 2924626915529, 109588276298995, 4456269669580742, 195418762093000328, 9192090435429906463, 461630086359185798777, 24651183861530752336994, 1394716088179233110318104
Offset: 0

Views

Author

Seiichi Manyama, Dec 24 2023

Keywords

Crossrefs

Cf. A291979, A368449.
Cf. A368453.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=1, i, 3^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 1 + Sum_{k=1..n} 3^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A368284 Expansion of e.g.f. exp(-2*x) / (1 + log(1 - x)).

Original entry on oeis.org

1, -1, 3, 0, 32, 182, 1882, 20500, 260136, 3701968, 58565360, 1019110848, 19346296752, 397867297136, 8811800026928, 209100451072672, 5292665533921024, 142338738348972672, 4053176346277660288, 121828547313861426176, 3854597854165079424768
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A007840, A291979, A330149, A368283.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-2)^i+sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = (-2)^n + Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).

A368444 Expansion of e.g.f. exp(x) / (1 + log(1 - 2*x)).

Original entry on oeis.org

1, 3, 17, 155, 1937, 30499, 577793, 12784155, 323427041, 9207390211, 291277318065, 10136705490779, 384848820035057, 15829002092015267, 701141988610115617, 33275461169171553371, 1684504951149122303169, 90604594879948059236099
Offset: 0

Views

Author

Seiichi Manyama, Dec 24 2023

Keywords

Crossrefs

Cf. A291979, A368445.
Cf. A343707, A368283.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=1, i, 2^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 1 + Sum_{k=1..n} 2^k * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ sqrt(Pi) * exp(1/2 - exp(-1)/2) * 2^(n + 1/2) * n^(n + 1/2) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Dec 25 2023

A368449 Expansion of e.g.f. exp(x) / (1 + log(1 - 2*x)/2).

Original entry on oeis.org

1, 2, 7, 42, 365, 4090, 55699, 890722, 16341849, 338128594, 7786397471, 197460558394, 5467207989957, 164085022299146, 5305738076252587, 183876885720455218, 6798985094507177137, 267160159254659407650, 11116956337133269707319, 488348854052875260086474
Offset: 0

Views

Author

Seiichi Manyama, Dec 24 2023

Keywords

Crossrefs

Cf. A291979, A368450.
Cf. A227917.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=1, i, 2^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 1 + Sum_{k=1..n} 2^(k-1) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-10 of 11 results. Next

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