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 1-3 of 3 results.

A385372 Expansion of e.g.f. 1/(1 - 3 * arcsinh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 27, 264, 3369, 52896, 986187, 21293184, 522491697, 14359993344, 436964488443, 14583637923840, 529683272760537, 20798444046458880, 877927319167721067, 39644175780617748480, 1906959640776766940385, 97344936393086594580480, 5255894631271228490720475
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2025

Keywords

Crossrefs

Cf. A296675, A385371.
Cf. A007559, A385311, A385343.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asinh(x))^(1/3)))

Formula

E.g.f.: 1/(1 - 3 * log(x + sqrt(x^2 + 1)))^(1/3).
a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (exp(2/3) + 1)^(1/3) * 2^(n + 1/2) * n^(n - 1/6) / (3^(1/3) * Gamma(1/3) * exp(2*n/3) * (exp(2/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Jun 27 2025

A385367 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x)).

Original entry on oeis.org

1, 2, 8, 46, 352, 3378, 38912, 522702, 8024064, 138586722, 2659565568, 56141737518, 1292851544064, 32253357421842, 866534937329664, 24943658876605902, 765883864848531456, 24985882009464388290, 863077992845681885184, 31469256501815056673070
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Crossrefs

Cf. A296675, A385368.
Cf. A007289, A385343, A385346, A385371.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-2ArcSinh[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 14 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*asinh(x))))

Formula

E.g.f.: 1/(1 - 2 * log(x + sqrt(x^2 + 1))).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A385371.
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (1 + exp(1)) * 2^(n - 1/2) * n^(n + 1/2) / ((exp(1) - 1)^(n+1) * exp(n/2)). - Vaclav Kotesovec, Jun 27 2025

A385419 Expansion of e.g.f. 1/(1 - arcsinh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 11, 57, 489, 5067, 50595, 573297, 9323985, 168823443, 2679252795, 45149256105, 1121782132665, 29930127386715, 629179051311315, 13329925622622945, 472248682257228705, 17395967794618282275, 434384524558247177835, 10095605146704332967705
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Comments

a(32) = -243211075187578815197768727974208613120575.

Crossrefs

Cf. A296675, A385420.
Cf. A001147, A385371, A385343.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-asinh(2*x))^(1/2)))

Formula

E.g.f.: 1/(1 - log(2*x + sqrt(4*x^2 + 1)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A385343(n,k), where i is the imaginary unit.
Showing 1-3 of 3 results.

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

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