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.

A367158 E.g.f. satisfies A(x) = 1 - A(x)^3 * log(1 - x).

Original entry on oeis.org

1, 1, 7, 92, 1824, 48804, 1649724, 67492872, 3243567552, 179139978072, 11181615816216, 778466939121552, 59811143359463952, 5027200928936108064, 458865351655379262432, 45201262487568977507328, 4779609140451030860102400, 539990133396500652971120640
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2023

Keywords

Crossrefs

Cf. A367155, A367161, A367164.
Cf. A052803.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k) * (3*k)!/(2*k+1)! * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 10 2023 *)
  • PARI
    a(n) = sum(k=0, n, (3*k)!/(2*k+1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (3*k)!/(2*k+1)! * |Stirling1(n,k)|.
a(n) ~ 9 * n^(n-1) / (2^(5/2) * exp(23*n/27) * (exp(4/27) - 1)^(n - 1/2)). - Vaclav Kotesovec, Nov 10 2023

A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

Original entry on oeis.org

1, 1, 8, 64, 824, 12968, 252720, 5789712, 153169440, 4589004192, 153643615872, 5684390364288, 230307823878144, 10141452865049088, 482259966649655808, 24630247225278881280, 1344614199041549399040, 78137673004382654223360
Offset: 1

Views

Author

Vladeta Jovovic, Oct 18 2003

Keywords

Links

Crossrefs

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287, A293604.
Cf. A370462, A370463.

Programs

  • Mathematica
    Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)
  • PARI
    x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ G. C. Greubel, May 24 2017

Formula

a(n) = Sum_{k=1..n} Stirling1(n, k)*k!*Catalan(k-1).
a(n) ~ n! / (2*exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (log(1 + x)) / (1 - A(x)).
E.g.f.: Series_Reversion( exp(x * (1 - x)) - 1 ). (End)

A087152 Expansion of (1-sqrt(1-4*log(1+x)))/log(1+x)/2.

Original entry on oeis.org

1, 3, 20, 194, 2554, 42226, 843744, 19769256, 531768120, 16152296424, 546895099200, 20425461026736, 834215500905552, 36988602430554576, 1769524998544143360, 90851799797294235264, 4982968503277896871296
Offset: 1

Views

Author

Vladeta Jovovic, Oct 18 2003

Keywords

Links

Crossrefs

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287.

Programs

  • Mathematica
    Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/Log[1+x]/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)
  • PARI
    x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/log(1+x)/2 - 1)) \\ G. C. Greubel, May 24 2017

Formula

a(n) = Sum_{k=0..n} Stirling1(n, k)*k!*Catalan(k).
a(n) ~ 2*n! / (exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015

A371140 E.g.f. satisfies A(x) = 1 - x*A(x)^2 * log(1 - x).

Original entry on oeis.org

1, 0, 2, 3, 56, 270, 5064, 47040, 984416, 14116032, 336538080, 6589416240, 179336461248, 4446985514400, 137520942168960, 4112410749501600, 143445512622458880, 5004065722611594240, 195260931334478223360, 7762385328551718796800, 336051947630616458065920
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2024

Keywords

Crossrefs

Cf. A052830, A371141.
Cf. A052803.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(-1 + Sqrt[1 + 4*x*Log[1 - x]])/(2*x*Log[1 - x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 12 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(2/(1+sqrt(1+4*x*log(1-x)))))
  • PARI
    a(n) = n!*sum(k=0, n\2, (2*k)!/(k+1)!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

E.g.f.: 2/(1 + sqrt(1+4*x*log(1-x))).
a(n) = n! * Sum_{k=0..floor(n/2)} (2*k)!/(k+1)! * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(2 + 8*r^2/(1-r)) * n^(n-1) / (exp(n) * r^n), where r = 0.436224579489690436773045325306926562580857950193340891933383996... is the root of the equation 4*r*log(1-r) = -1. - Vaclav Kotesovec, Mar 12 2024

A377445 E.g.f. satisfies A(x) = 1/(1 + A(x) * log(1 - x))^2.

Original entry on oeis.org

1, 2, 16, 226, 4678, 128728, 4437416, 184176816, 8949477600, 498611374704, 31343763192144, 2194986671431200, 169478318264408832, 14304849733469090976, 1310439414650613267552, 129495512412669053694720, 13731040497246647099309568, 1555129289690056322821075968
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A052803, A377446, A377449.
Cf. A367158.

Programs

  • Mathematica
    Table[2*Sum[(3*k+1)!/(2*k+2)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2025 *)
  • PARI
    a(n) = 2*sum(k=0, n, (3*k+1)!/(2*k+2)!*abs(stirling(n, k, 1)));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367158.
a(n) = 2 * Sum_{k=0..n} (3*k+1)!/(2*k+2)! * |Stirling1(n,k)|.
a(n) ~ 27 * n^(n-1) / (2^(5/2) * (exp(4/27) - 1)^(n - 1/2) * exp(23*n/27)). - Vaclav Kotesovec, Aug 27 2025

A377446 E.g.f. satisfies A(x) = 1/(1 + A(x) * log(1 - x))^3.

Original entry on oeis.org

1, 3, 33, 642, 18312, 694242, 32960910, 1883757264, 126015816624, 9664169177136, 836144408644560, 80584887639709296, 8562470160920850144, 994509363776066618256, 125361725299037966371824, 17045520282440541500805504, 2486876242354800277464657792, 387527092638347603108279296512
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A052803, A377445, A377449.
Cf. A377447.

Programs

  • PARI
    a(n) = 3*sum(k=0, n, (4*k+2)!/(3*k+3)!*abs(stirling(n, k, 1)));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A377447.
a(n) = 3 * Sum_{k=0..n} (4*k+2)!/(3*k+3)! * |Stirling1(n,k)|.

A377449 E.g.f. satisfies A(x) = 1/(1 + A(x) * log(1 - x))^4.

Original entry on oeis.org

1, 4, 56, 1388, 50444, 2436176, 147308248, 10720410984, 913099165080, 89150817350880, 9819313409197632, 1204676163038931744, 162935364815509750368, 24088567621306193343360, 3864931159784777490964608, 668886871993798772730203136, 124209455281616641852564586496
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A052803, A377445, A377446.
Cf. A377448.

Programs

  • PARI
    a(n) = 4*sum(k=0, n, (5*k+3)!/(4*k+4)!*abs(stirling(n, k, 1)));

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377448.
a(n) = 4 * Sum_{k=0..n} (5*k+3)!/(4*k+4)! * |Stirling1(n,k)|.

A377692 E.g.f. satisfies A(x) = (1 - log(1 - x) * A(x))^2.

Original entry on oeis.org

1, 2, 12, 118, 1634, 29408, 654040, 17362056, 536410200, 18922946928, 750902659200, 33118793900784, 1607673329621712, 85192554602094912, 4894219487974911552, 303021216528999244416, 20116223556200658052992, 1425479651299747192856832, 107400336067263661850548224
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2024

Keywords

Crossrefs

Cf. A007840, A377693.
Cf. A052803, A377445.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (2*k+1)!/(k+2)!*abs(stirling(n, k, 1)));

Formula

E.g.f.: 4/(1 + sqrt(1 + 4*log(1-x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052803.
a(n) = 2 * Sum_{k=0..n} (2*k+1)!/(k+2)! * |Stirling1(n,k)|.
a(n) ~ 2^(7/2) * n^(n-1) / ((exp(1/4) - 1)^(n - 1/2) * exp(3*n/4)). - Vaclav Kotesovec, Aug 27 2025
Showing 1-8 of 8 results.

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