A133867 -id:A133867 - cp's OEIS frontend

A201946 Decimal expansion of x>0 satisfying x*sinh(x)=2.

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

Offset: 1

Clark Kimberling, Dec 15 2011

For many choices of u and v, there is exactly one x>0 satisfying x*sinh(u*x)=v.  Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A133867
1.... 2.... A201946
1.... 3.... A202243
2.... 1.... A202244
3.... 1.... A202245
2.... 2.... A202284
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 A199597, take f(x,u,v)=x*sinh(ux)-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.

			1.2493943366463244725112743212610081234694...

Index entries for transcendental numbers.

Cf. A201939.

(* Program 1:  A201946 *)
u = 1; v = 2;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201946 *)
(* Program 2: implicit surface of u*sinh(x)=v *)
f[{x_, u_, v_}] := x*Sinh[u*x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 10}, {u, 1, 4}];
ListPlot3D[Flatten[t, 1]] (* for A201946 *)

A009341 Expansion of e.g.f. log(1 + sin(x)*x), even powers only.

Original entry on oeis.org

0, 2, -16, 366, -17704, 1467370, -185815884, 33370050910, -8067253019536, 2526062494781394, -994534162338738580, 480859837194669214150, -280103496938395910686680, 193472520727526106582807226

Offset: 0

R. H. Hardin

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A133867.

With[{nn=30},Take[CoefficientList[Series[Log[1+Sin[x]x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Nov 27 2013 *)

a(n):=2*sum(binomial(2*n,k)*(k-1)!*(sum((2*i-k)^(2*n-k)*binomial(k,i)*(-1)^(n-i+k-1),i,0,k/2))/(2^k),k,1,2*n-1); /* Vladimir Kruchinin, Jun 28 2011 */

a(n) = 2*sum(k=1..2*n-1, binomial(2*n,k)*(k-1)!*(sum(i=0..k/2, (2*i-k)^(2*n-k)*binomial(k,i)*(-1)^(n-i+k-1)))/(2^k)). - Vladimir Kruchinin, Jun 28 2011

a(n) ~ (-1)^(n+1) * (2*n)! / (n*r^(2*n)), where r = 0.9320200293523439... (see A133867) is the root of the equation r*sinh(r)=1. - Vaclav Kotesovec, Apr 20 2014

Extended with signs by Olivier Gérard, Mar 15 1997

Previous Mathematica program replaced by Harvey P. Dale, Nov 27 2013

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A201946 Decimal expansion of x>0 satisfying x*sinh(x)=2.

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