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.

A069814 Decimal expansion of root of the equation x*cosh(x)=1.

Original entry on oeis.org

7, 6, 5, 0, 0, 9, 9, 5, 4, 5, 5, 0, 7, 3, 2, 1, 2, 2, 6, 5, 5, 3, 2, 1, 7, 4, 2, 4, 8, 2, 8, 1, 5, 2, 1, 9, 2, 0, 0, 3, 5, 2, 1, 3, 7, 4, 7, 5, 0, 4, 3, 3, 2, 3, 1, 6, 2, 4, 7, 0, 7, 1, 0, 7, 4, 0, 0, 1, 9, 2, 3, 1, 9, 9, 4, 4, 8, 1, 4, 1, 2, 7, 8, 0, 8, 2
Offset: 0

Views

Author

Benoit Cloitre, Apr 30 2002

Keywords

Examples

			0.76500995455073212265532174248281521920035213747504332316247071...

Links

Crossrefs

Cf. A205571, A385281, A385282.

Programs

  • Maple
    Digits:= 120:
fsolve(x*cosh(x)=1,x=0..1); # Robert Israel, Jan 07 2015
  • Mathematica
    RealDigits[x/.FindRoot[x*Cosh[x]==1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 06 2015 *)
  • PARI
    solve(x=0, 1, x*cosh(x)-1) \\ Michel Marcus, Jan 06 2015

Extensions

Corrected by Harvey P. Dale, Jan 06 2015

A385308 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 18, 141, 1400, 17055, 245392, 4070073, 76483584, 1606033755, 37267953536, 947051118981, 26156846230528, 780174007426359, 24992424003517440, 855795857724702705, 31193844533488074752, 1205893835653392258867, 49280187764171870470144, 2122704756621224015194365
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385309.
Cf. A380015, A385304, A385306, A385310.
Cf. A001147, A185951, A352648, A385281.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ sqrt(2) * n^n / (sqrt(1 + r*sqrt(1 - 4*r^2)) * exp(n) * r^n), where r = 0.452787214835453627588998503316635625709288535855... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Jun 28 2025

A385282 Expansion of e.g.f. 1/(1 - 3 * x * cosh(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 55, 712, 11605, 248320, 6218443, 178519936, 5846857993, 214490045440, 8700546508159, 387053184719872, 18737207168958109, 980424546959183872, 55142056940797803475, 3317502712746788945920, 212592531182720568805777, 14456626429227650204041216
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385281.
Cf. A007559, A185951.
Cf. A069814.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*3^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * (1/r + sqrt(1 - r^2))^(1/3) * exp(n) * r^(n + 1/3)), where r = A069814. - Vaclav Kotesovec, Jun 24 2025

A385283 Expansion of e.g.f. 1/(1 - 2 * x * cos(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 3, -39, -775, -9045, -85813, -426447, 7321329, 325555155, 7786757011, 137053423881, 1388713844713, -21121997539461, -1827406866674085, -69034283067822495, -1852635543265039903, -30574875232261547613, 308376017794648053539, 54871741689019890859065
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385284.
Cf. A001586, A235134, A380155, A385281.
Cf. A001147, A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*(2*I)^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.
Showing 1-4 of 4 results.

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