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 1-8 of 8 results.

A320343 Expansion of e.g.f. 1/sqrt(1 - 2*log(1 + x)).

Original entry on oeis.org

1, 1, 2, 8, 42, 294, 2472, 24828, 286164, 3751428, 54864408, 887989200, 15731200680, 303068103480, 6304498706880, 140890167340560, 3365469544248720, 85585469309951760, 2308349518803845280, 65819488298810181120, 1978202007765686904480, 62505106242073569018720, 2071320752120227622985600
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2019

Keywords

Crossrefs

Cf. A001147, A006252, A048994, A088501, A305404, A346978.

Programs

  • Maple
    seq(n!*coeff(series(1/sqrt(1-2*log(1+x)),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 29 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[1/Sqrt[1 - 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A001147(k).
a(n) ~ n^n / ((exp(1/2) - 1)^(n + 1/2) * exp(n - 1/4)). - Vaclav Kotesovec, Jan 29 2019
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 - k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A346987 Expansion of e.g.f. 1 / (1 + 5 * log(1 - x))^(1/5).

Original entry on oeis.org

1, 1, 7, 86, 1524, 35370, 1015590, 34757400, 1381147440, 62498177880, 3172764322680, 178566159846480, 11034757650750960, 742773843654742080, 54094804600076176320, 4238009228531321452800, 355400361455423327193600, 31764402860426288679456000, 3014207878695233997923193600
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Links

Crossrefs

Cf. A007840, A008548, A346978, A346984, A347015, A347016, A347019.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 + 5 Log[1 - x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]
  • Maxima
    a[n]:=if n=0 then 1 else sum(n!/(n-k)!*(5/k-4/n)*a[n-k],k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Aug 27 2023 */

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A008548(k).
a(n) ~ n! * exp(n/5) / (Gamma(1/5) * 5^(1/5) * n^(4/5) * (exp(1/5) - 1)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
For n > 0, a(n) = Sum_{k=1..n} (n!/(n-k)!)*(5/k-4/n)*a(n-k). - Tani Akinari, Aug 27 2023

A347015 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(1/3).

Original entry on oeis.org

1, 1, 5, 42, 498, 7644, 144156, 3225648, 83536008, 2457701928, 80970232104, 2953056534768, 118112744060208, 5140622709134496, 241863782829704928, 12232551538417012992, 661818290353375962240, 38140594162828447248000, 2332567001993176540206720, 150880256846462633823648000
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A007559, A007840, A346978, A346982, A347016.
Cf. A354263.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
seq(a(n), n=0..19);  # Alois P. Heinz, Aug 10 2021
  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 + 3 Log[1 - x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007559(k).
a(n) ~ n! * exp(n/3) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A347016 Expansion of e.g.f. 1 / (1 + 4 * log(1 - x))^(1/4).

Original entry on oeis.org

1, 1, 6, 62, 916, 17644, 419360, 11859840, 388965600, 14514046560, 607165485120, 28143329181120, 1431690475207680, 79302863940387840, 4751108622148907520, 306118435580577146880, 21107196651940518551040, 1550773243761690603179520, 120947288498720390755353600
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A007696, A007840, A346978, A346983, A347015.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021
  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 + 4 Log[1 - x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007696(k).
a(n) ~ n! * exp(n/4) / (Gamma(1/4) * 2^(1/2) * n^(3/4) * (exp(1/4) - 1)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A347019 E.g.f.: 1 / (1 + 6 * log(1 - x))^(1/6).

Original entry on oeis.org

1, 1, 8, 114, 2358, 64074, 2157828, 86714592, 4049302404, 215458069428, 12867377875632, 852254389954296, 61998666080311800, 4914000741835488744, 421488717980664846960, 38897664480760253351904, 3843081247426270376211216, 404727487161912602921083536
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Comments

In general, for k >= 1, if e.g.f. = 1 / (1 + k*log(1 - x))^(1/k), then a(n) ~ n! * exp(n/k) / (Gamma(1/k) * k^(1/k) * n^(1 - 1/k) * (exp(1/k) - 1)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Links

Crossrefs

Cf. A007840, A008542, A346978, A346985, A346987, A347015, A347016.

Programs

  • Mathematica
    nmax = 17; CoefficientList[Series[1/(1 + 6 Log[1 - x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A008542(k).
a(n) ~ n! * exp(n/6) / (Gamma(1/6) * 6^(1/6) * n^(5/6) * (exp(1/6) - 1)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021

A354261 Expansion of e.g.f. 1/sqrt(1 + 6 * log(1-x)).

Original entry on oeis.org

1, 3, 30, 492, 11250, 330282, 11844288, 501822108, 24527880756, 1358556883308, 84094256900232, 5753027212816320, 431039748845205000, 35102411472973316040, 3087236653107610062240, 291627772873980244894800, 29447260745861893561906320
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A346978, A354241, A354262.
Cf. A354252, A354259.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+6*log(1-x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(-3*log(1-x)/2)^k)))
  • PARI
    a(n) = sum(k=0, n, (3/2)^k*(2*k)!*abs(stirling(n, k, 1))/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (-3 * log(1-x)/2)^k.
a(n) = Sum_{k=0..n} (3/2)^k * (2*k)! * |Stirling1(n,k)|/k!.
a(n) ~ n^n / (sqrt(3) * (exp(1/6)-1)^(n + 1/2) * exp(5*n/6)). - Vaclav Kotesovec, Jun 04 2022

A354262 Expansion of e.g.f. 1/sqrt(1 + 8 * log(1-x)).

Original entry on oeis.org

1, 4, 52, 1112, 33192, 1272576, 59607552, 3298935552, 210638509824, 15241340093952, 1232504690492928, 110154484622208000, 10782300230031713280, 1147157496053856645120, 131810751499551281786880, 16266976762439018716323840, 2145960434809665656603320320
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A346978, A354241, A354261.
Cf. A354253, A354260.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/Sqrt[1+8*Log[1-x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 14 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+8*log(1-x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(-2*log(1-x))^k)))
  • PARI
    a(n) = sum(k=0, n, 2^k*(2*k)!*abs(stirling(n, k, 1))/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (-2 * log(1-x))^k.
a(n) = Sum_{k=0..n} 2^k * (2*k)! * |Stirling1(n,k)|/k!.
a(n) ~ n^n / (2 * (exp(1/8)-1)^(n + 1/2) * exp(7*n/8)). - Vaclav Kotesovec, Jun 04 2022

A375687 Expansion of e.g.f. 1 / sqrt(1 + 2 * x * log(1 - x)).

Original entry on oeis.org

1, 0, 2, 3, 44, 210, 2934, 26040, 404592, 5302584, 95029560, 1632252600, 33865401096, 712672337520, 16986980278800, 420485947572600, 11386595338156800, 322890555922925760, 9820815078397642560, 313247186941438569600, 10588974153880701225600
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Cf. A001147, A052830, A346978, A367878.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+2*x*log(1-x))))
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = n!*sum(k=0, n, a001147(k)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} A001147(k) * |Stirling1(n-k,k)|/(n-k)!.
Showing 1-8 of 8 results.

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