A051397 -id:A051397 - cp's OEIS frontend

A073742 Decimal expansion of sinh(1).

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

Offset: 1

Rick L. Shepherd, Aug 07 2002

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

Decimal expansion of u > 0 such that 1 = arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (u,y(u)). - Clark Kimberling, Jul 04 2020

			1.17520119364380145688238185059...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:7 at page 20.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068139 (continued fraction), A073743, A073744, A073745, A073746, A073747, A049469, A049470, A174548.

First@ RealDigits@ N[Sinh@ 1, 120] (* Michael De Vlieger, Sep 04 2016 *)

PARI
```
sinh(1)
```

Equals (e - e^(-1))/2.
Equals sin(i)/i. - N. J. A. Sloane, Feb 12 2010
Equals Sum_{n>=0} 1/A009445(n). See Gradsteyn-Ryzhik (0.245.6.) - R. J. Mathar, Oct 27 2012
Continued fraction representation: sinh(1) = 1 + 1/(6 - 6/(21 - 20/(43 - 42/(73 - ... - (2*n - 1)*(2*n - 2)/((2*n*(2*n + 1) + 1) - ... ))))). See A051397 for proof. Cf. A049469. - Peter Bala, Sep 02 2016
Equals Product_{k>=1} 1 + 1/(k * Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073745 = A174548/2. - Hugo Pfoertner, Dec 27 2024

A009628 Expansion of e.g.f.: sinh(x)/(1+x).

Original entry on oeis.org

0, 1, -2, 7, -28, 141, -846, 5923, -47384, 426457, -4264570, 46910271, -562923252, 7318002277, -102452031878, 1536780478171, -24588487650736, 418004290062513, -7524077221125234, 142957467201379447, -2859149344027588940, 60042136224579367741

Offset: 0

R. H. Hardin

(-1)^n*(A000166 + A000522)/2 = A009179, (-1)^n*(A000166-A000522)/2 = this_sequence.

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000166, A000522, A009179, A051397, A087208, A186763.

G(x):= sinh(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

a[n_] := (-1)^n (Exp[-1] Gamma[1 + n, -1] - Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 18 2017 *)
With[{nn=30},CoefficientList[Series[Sinh[x]/(1+x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 19 2023 *)

a(n) = n!*polcoeff((sinh(x)/(1+x) + x * O(x^n)), n) \\ Charles R Greathouse IV, Sep 09 2016

x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)/(1+x)))) \\ Altug Alkan, Dec 18 2017

def A009628(n)
  a = 0
  (0..n).map{|i| a = -i * a + i % 2}
end # Seiichi Manyama, Sep 09 2016

a(n) = (-1)^(n+1)*floor(n!*sinh(1)), n>=1. - Vladeta Jovovic, Aug 10 2002
Let u(1) = 1, u(n) = n*u(n-1) + n (mod 2); then for n>0, a(n) = (-1)^(n+1)*u(n). - Benoit Cloitre, Jan 12 2003
Unsigned sequence satisfies a(n) = n*a(n-1)+a(n-2)-(n-2)*a(n-3), with E.g.f. sinh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 08 2003
a(n) = (-1)^(n+1) * n! * Sum_{k=1..floor((n+1)/2)} 1/(2*k-1)!.
a(n) = -n*a(n-1) + n (mod 2). - Seiichi Manyama, Sep 09 2016
a(n) = (-1)^n*(exp(-1)*Gamma(1+n,-1) - exp(1)*Gamma(1+n,1))/2. - Peter Luschny, Dec 18 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Mar 19 2023

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

Original entry on oeis.org

1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100089, A143820, A330044, A337726, A337727, A337728, A337729, A337730.

Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 1)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Pi/6 - Sqrt[3] x/2])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 1}], {n, 0, 11}]
Table[Floor[(Exp[3/2] + 2 Sin[(3 Sqrt[3] - Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 1)!], {n, 0, 11}]

a(n) = (3*n+1)!*sum(k=0, n, 1/(3*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(Pi/6 - sqrt(3)*x/2)) / (3*exp(x/2) * (1 - x^3)) = x + 25*x^4/4! + 5251*x^7/7! + 3780721*x^10/10! + ...

a(n) = floor(c * (3*n+1)!), where c = (exp(3/2) + 2 * sin((3 * sqrt(3) - Pi) / 6))/(3 * sqrt(exp(1))) = A143820.

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

Original entry on oeis.org

1, 25, 42001, 498971881, 21795091762081, 2534333270094778681, 646315807872650838343345, 317599587988620621961919733001, 274101148417699141578015206369183041, 387502275541069630431671657548241448722521, 849931991080760484603611346800010863970028660561

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100733, A330045, A332890, A337725, A337726, A337728, A337729, A337730.

Table[(4 n)! Sum[1/(4 k)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n)! SeriesCoefficient[(1/2) (Cos[x] + Cosh[x])/(1 - x^4), {x, 0, 4 n}], {n, 0, 10}]
Table[Floor[(1/2) (Cos[1] + Cosh[1]) (4 n)!], {n, 0, 10}]

a(n) = (4*n)!*sum(k=0, n, 1/(4*k)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cos(x) + cosh(x)) / (1 - x^4) = 1 + 25*x^4/4! + 42001*x^8/8! + 498971881*x^12/12! + ...

a(n) = floor(c * (4*n)!), where c = (cos(1) + cosh(1)) / 2 = A332890.

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

Original entry on oeis.org

1, 121, 365905, 6278929801, 358652470233121, 51516840824285500441, 15640512874253077933887601, 8915467710633236496186345872425, 8755702529258688898174686554391144001, 13878488965077362598718732163634314533105081, 33731389859841228248933904149069928786421237268881

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334363, A337725, A337726, A337727, A337729, A337730.

Table[(4 n + 1)! Sum[1/(4 k + 1)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 1)! SeriesCoefficient[(1/2) (Sin[x] + Sinh[x])/(1 - x^4), {x, 0, 4 n + 1}], {n, 0, 10}]
Table[Floor[(1/2) (Sin[1] + Sinh[1]) (4 n + 1)!], {n, 0, 10}]

a(n) = (4*n+1)!*sum(k=0, n, 1/(4*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (sin(x) + sinh(x)) / (1 - x^4) = x + 121*x^5/5! + 365905*x^9/9! + 6278929801*x^13/13! + ...

a(n) = floor(c * (4*n+1)!), where c = (sin(1) + sinh(1)) / 2 = A334363.

A337726 a(n) = (3n+2)! Sum_{k=0..n} 1 / (3*k+2)!.

Original entry on oeis.org

1, 61, 20497, 20292031, 44317795705, 180816606476401, 1236785588298582841, 13142083661260741268467, 205016505115667563788085201, 4494781858155895668489979946725, 133764708098719455094261803214536001, 5252940087036713001551661012234828759271

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100043, A143821, A330044, A337725, A337727, A337728, A337729, A337730.

Table[(3 n + 2)! Sum[1/(3 k + 2)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 2)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Sqrt[3] x/2 + Pi/6])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 2}], {n, 0, 11}]
Table[Floor[(Exp[3/2] - 2 Sin[(3 Sqrt[3] + Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 2)!], {n, 0, 11}]

a(n) = (3*n+2)!*sum(k=0, n, 1/(3*k+2)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(sqrt(3)*x/2 + Pi/6)) / (3*exp(x/2) * (1 - x^3)) = x^2/2! + 61*x^5/5! + 20497*x^8/8! + 20292031*x^11/11! + ...

a(n) = floor(c * (3*n+2)!), where c = (exp(3/2) - 2 * sin((3 * sqrt(3) + Pi) / 6))/(3 * sqrt(exp(1))) = A143821.

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.

Original entry on oeis.org

1, 361, 1819441, 43710250585, 3210080802962401, 563561785768079119561, 202205968733586788098486801, 132994909755454702268136738753721, 148026526435655214537290625514621562305, 262237873172349351865682580536682974917045801, 704454843460345510903820429747302209179158476142321

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334364, A337725, A337726, A337727, A337728, A337730.

Table[(4 n + 2)! Sum[1/(4 k + 2)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 2)! SeriesCoefficient[(1/2) (Cosh[x] - Cos[x])/(1 - x^4), {x, 0, 4 n + 2}], {n, 0, 10}]
Table[Floor[(1/2) (Cosh[1] - Cos[1]) (4 n + 2)!], {n, 0, 10}]

a(n) = (4*n+2)!*sum(k=0, n, 1/(4*k+2)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cosh(x) - cos(x)) / (1 - x^4) = x^2/2! + 361*x^6/6! + 1819441*x^10/10! + 43710250585*x^14/14! + ...

a(n) = floor(c * (4*n+2)!), where c = (cosh(1) - cos(1)) / 2 = A334364.

A337730 a(n) = (4n+3)! Sum_{k=0..n} 1 / (4*k+3)!.

Original entry on oeis.org

1, 841, 6660721, 218205219961, 20298322381652065, 4313799472548696853801, 1816972337837511114820981201, 1372104830641374893468212163747161, 1724241814377177346127894133451232399041, 3403694723384093133512770088891935585284510985

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334365, A337725, A337726, A337727, A337728, A337729.

Table[(4 n + 3)! Sum[1/(4 k + 3)!, {k, 0, n}], {n, 0, 9}]
Table[(4 n + 3)! SeriesCoefficient[(1/2) (Sinh[x] - Sin[x])/(1 - x^4), {x, 0, 4 n + 3}], {n, 0, 9}]
Table[Floor[(1/2) (Sinh[1] - Sin[1]) (4 n + 3)!], {n, 0, 9}]

a(n) = (4*n+3)!*sum(k=0, n, 1/(4*k+3)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (sinh(x) - sin(x)) / (1 - x^4) = x^3/3! + 841*x^7/7! + 6660721*x^11/11! + 218205219961*x^15/15! + ...

a(n) = floor(c * (4*n+3)!), where c = (sinh(1) - sin(1)) / 2 = A334365.

A074790 a(n) = (2n+1)!Sum_{k=0..n} (-1)^k/(2*k+1)!.

Original entry on oeis.org

1, 5, 101, 4241, 305353, 33588829, 5239857325, 1100370038249, 299300650403729, 102360822438075317, 42991545423991633141, 21753721984539766369345, 13052233190723859821607001, 9162667699888149594768114701

Offset: 0

Benoit Cloitre, Sep 07 2002

Seiichi Manyama, Table of n, a(n) for n = 0..224

Cf. A000166, A009551, A049469, A051397.

Table[(2n+1)!Sum[(-1)^k/(2k+1)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 14 2019 *)

a(n) = (2*n+1)!*sum(k=0,n,(-1)^k/(2*k+1)!); \\ Michel Marcus, Sep 09 2016

[factorial(2*n+1)*sum((-1)^j/factorial(2*j+1) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 09 2021

a(n) = round(sin(1)*(2*n+1)!).
a(n) = A009551(2*n+1).
From Peter Bala, Jan 30 2015: (Start)
G.f.: sin(x)/(1 - x^2) = x + 5*x^3/3! + 101*x^5/5! + 4241*x^7/7! + ....
a(n) = 2*n*(2*n + 1)*a(n-1) + (-1)^n with a(0) = 1.
a(n) = (4*n^2 + 2*n - 1)*a(n-1) + (2*n-1)*(2*n-2)*a(n-2) with a(0) = 1, a(1) = 5.
The sequence b(n) := (2*n + 1)! also satisfies the second recurrence but with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = (2*n + 1)!*(1 - 1/(6 + 6/(19 + 20/(41 + ... + (2*n - 1)*(2*n - 2)/(4*n^2 + 2*n - 1) )))) for n >= 2. Taking the limit gives the continued fraction representation sin(1) = 1 - 1/(6 + 6/(19 + 20/(41 + ... + (2*n - 1)*(2*n - 2)/((4*n^2 + 2*n - 1) + ... )))). (End)

A195326 Numerators of fractions leading to e - 1/e (A174548).

Original entry on oeis.org

0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447

Offset: 0

Paul Curtz, Oct 12 2011

The sequence of approximations of exp(1) obtained by truncating the Taylor series of exp(x) after n terms is A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, ...
A Taylor series of exp(-1) is 1, 0, 1/2, 1/3, 3/8, ... and (apart from the first 2 terms) given by A000255(n)/A001048(n). Subtracting both sequences term by term we obtain a series for exp(1) - exp(-1) = 0, 2, 2, 7/3, 7/3, 47/20, 47/20, 5923/2520, 5923/2520, 426457/181440, 426457/181440, ... which defines the numerators here.
Each second of the denominators (that is 3, 2520, 19958400, ...) is found in A085990 (where each third term, that is 60, 19958400, ...) is to be omitted.
This numerator sequence here is basically obtained by doubling entries of A051397, A009628, A087208, or A186763, caused by the standard associations between cosh(x), sinh(x) and exp(x).

			a(0) =  1  -  1;
a(1) =  2  -  0;
a(2) = 5/2 - 1/2.

Cf. A001113, A068985, A197222, A197223.

taylExp1 := proc(n)
        add(1/j!,j=0..n) ;
end proc:
A000255 := proc(n)
        if n <=1 then
                1;
        else
                n*procname(n-1)+(n-1)*procname(n-2) ;
        end if;
end proc:
A001048 := proc(n)
        n!+(n-1)! ;
end proc:
A195326 := proc(n)
        if n = 0 then
                0;
        elif n =1 then
                2;
        else
                taylExp1(n) -A000255(n-2)/A001048(n-1);
        end if;
          numer(%);
end proc:
seq(A195326(n),n=0..20) ; # R. J. Mathar, Oct 14 2011

Material meant to be placed in other sequences removed by R. J. Mathar, Oct 14 2011

cp's OEIS Frontend

A073742 Decimal expansion of sinh(1).

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A009628 Expansion of e.g.f.: sinh(x)/(1+x).

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A337725 a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.

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A337727 a(n) = (4*n)! * Sum_{k=0..n} 1 / (4*k)!.

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A337728 a(n) = (4*n+1)! * Sum_{k=0..n} 1 / (4*k+1)!.

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A337726 a(n) = (3*n+2)! * Sum_{k=0..n} 1 / (3*k+2)!.

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A337729 a(n) = (4*n+2)! * Sum_{k=0..n} 1 / (4*k+2)!.

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A337730 a(n) = (4*n+3)! * Sum_{k=0..n} 1 / (4*k+3)!.

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A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

A337726 a(n) = (3n+2)! Sum_{k=0..n} 1 / (3*k+2)!.

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.

A337730 a(n) = (4n+3)! Sum_{k=0..n} 1 / (4*k+3)!.

A074790 a(n) = (2n+1)!Sum_{k=0..n} (-1)^k/(2*k+1)!.