A069814 -id:A069814 - cp's OEIS frontend

A205571 Expansion of e.g.f. 1/(1 - x*cosh(x)).

1, 1, 2, 9, 48, 305, 2400, 22057, 230272, 2708001, 35412480, 509177801, 7986468864, 135718942801, 2483729876992, 48699677975145, 1018542257111040, 22634000289407297, 532557637644976128, 13226748101381102473, 345792863300174479360, 9492229607399841038961

Offset: 0

Paul D. Hanna, Jan 28 2012

nonn

Radius of convergence of e.g.f. is |x| < r where r = 0.7650099545507... satisfies cosh(r) = 1/r. See A069814.

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 305*x^5/5! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A069814, A185951.
Cf. A000111, A006154, A352252.
Cf. A381206, A381207.

CoefficientList[Series[1/(1-x*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2013 *)

{a(n)=n!*polcoeff(1/(1-x*cosh(x +x*O(x^n))),n)}

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

a(2*n-1) == 1 (mod 4), a(2*n+2) == 0 (mod 4), for n>=1.
a(n) ~ n!/(1+r*sqrt(1-r^2))*(1/r)^n, where r = A069814 = 0.7650099545507321... is the root of the equation r*cosh(r)=1. - Vaclav Kotesovec, Feb 13 2013
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
a(n) = Sum_{k=0..n} k! * A185951(n,k). - Seiichi Manyama, Feb 17 2025

A201939 Decimal expansion of x>0 satisfying x*cosh(x)=2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 15 2011

For many choices of u and v, there is exactly one x>0 satisfying x*cosh(u*x)=v.  Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A069814
1.... 2.... A201939
1.... 3.... A201943
2.... 1.... A201944
3.... 1.... A201945
2.... 2.... A202283
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A201939, take f(x,u,v)=x*cosh(u*x)-v and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			1.15058496741866394953493373361378819576...

Index entries for transcendental numbers.

Cf. A201946.

(* Program 1:  A201939 *)
u = 1; v = 2;
f[x_] := x*Cosh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A201939 *)
(* Program 2: implicit surface of u*cosh(x)=v *)
f[{x_, u_, v_}] := x*Cosh[u*x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 20}, {u, 1, 9}];
ListPlot3D[Flatten[t, 1]] (* for A201939 *)

A385281 Expansion of e.g.f. 1/(1 - 2 * x * cosh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 27, 249, 2825, 41355, 708883, 13888497, 309267729, 7698772755, 211585744139, 6367841422569, 208299923870233, 7357493992966299, 279095125351544835, 11316313498670411745, 488403056864943302177, 22355228989851909617187, 1081663315375339026249211

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A205571, A385282.
Cf. A001586, A235134, A380155, A385283.
Cf. A001147, A185951.
Cf. A069814.

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^(n-k)*a185951(n, k));

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A185951(n,k), where A185951(n,0) = 0^n.

a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + r*sqrt(1 - r^2)) * exp(n) * r^n), where r = A069814. - Vaclav Kotesovec, Jun 24 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

Seiichi Manyama, Jun 24 2025

nonn

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

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));

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

A009305 Expansion of e.g.f. log(1 + x*cosh(x)).

Original entry on oeis.org

0, 1, -1, 5, -18, 89, -600, 4717, -42896, 449073, -5287680, 69090581, -993391872, 15583801609, -264816161792, 4846181282685, -95022445824000, 1987373846425697, -44163232640630784, 1039121484066627877, -25807915421845422080, 674707915373741222841

Offset: 0

R. H. Hardin

Cf. A069814.

With[{nn=30},CoefficientList[Series[Log[1+Cosh[x]*x],{x,0,nn}],x]Range[ 0,nn]!] (* Harvey P. Dale, Mar 09 2013 *)

a(n):=n!*sum(sum((k-2*i)^(n-k)*binomial(k,i),i,0,k)/(2^k*(n-k)!)*(-1)^(k-1)/k,k,1,n-1)+(-1)^(n-1)*(n-1)!; /* Vladimir Kruchinin, Apr 21 2011 */

x='x+O('x^66); /* that many terms */
egf=log(1+x*cosh(x)); /* = x - 1/2*x^2 + 5/6*x^3 - 3/4*x^4 + 89/120*x^5 +-... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 21 2011 */

a(n) = n!*Sum_{k=1..n-1} ((Sum_{i=0..k} (k-2*i)^(n-k)*binomial(k,i)) /(2^k*(n-k)!)*(-1)^(k-1)/k) + (-1)^(n-1)*(n-1)!. - Vladimir Kruchinin, Apr 21 2011

a(n) ~ (n-1)! * (-1)^(n+1) / r^n, where r = 0.765009954550732122655321742482815219200352137475... (see A069814) is the root of the equation r*cosh(r) = 1 . - Vaclav Kotesovec, Jan 24 2015

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 21 2011

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A205571 Expansion of e.g.f. 1/(1 - x*cosh(x)).

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A201939 Decimal expansion of x>0 satisfying x*cosh(x)=2.

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A385281 Expansion of e.g.f. 1/(1 - 2 * x * cosh(2*x))^(1/2).

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

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A009305 Expansion of e.g.f. log(1 + x*cosh(x)).

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