A174548 -id:A174548 - cp's OEIS frontend

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

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

A073742 Decimal expansion of sinh(1).

Original entry on oeis.org

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

A052558 a(n) = n! ((-1)^n + 2n + 3)/4.

Original entry on oeis.org

1, 1, 4, 12, 72, 360, 2880, 20160, 201600, 1814400, 21772800, 239500800, 3353011200, 43589145600, 697426329600, 10461394944000, 188305108992000, 3201186852864000, 64023737057280000, 1216451004088320000, 26761922089943040000, 562000363888803840000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Stirling transform of (-1)^(n+1)*a(n-1) = [1, -1, 4, -12, 72, -360, ...] is A052841(n-1) = [1,0,2,6,38,270,...]. - Michael Somos, Mar 04 2004
The Stirling transform of this sequence is A258369. - Philippe Deléham, May 17 2005; corrected by Ilya Gutkovskiy, Jul 25 2018
Ignoring reflections, this is the number of ways of connecting n+2 equally-spaced points on a circle with a path of n+1 line segments. See A030077 for the number of distinct lengths. - T. D. Noe, Jan 05 2007
From Gary W. Adamson, Apr 20 2009: (Start)
Signed: (+ - - + + - - + +, ...) = eigensequence of triangle A002260.
Example: -360 = (1, 1, -1, -4, 12, 71) dot (1, -2, 3, -4, 5, -6) = (1, -2, -3, 16, 60, -432). (End)
a(n) is the number of odd fixed points in all permutations of {1, 2, ..., n+1}, Example: a(2)=4 because we have 1'23', 1'32, 312, 213', 231, and 321, where the odd fixed points are marked. - Emeric Deutsch, Jul 18 2009
a(n) is also the number of permutations of [n+1] starting with an even number. - Olivier Gérard, Nov 07 2011

T. D. Noe, Table of n, a(n) for n=0..100
A. Moreno Canadas, P. F. Fernandez Espinoza, and I. D. M. Gaviria, Categorification of some integer sequences via Kronecker Modules, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 500.

Cf. A174548, A052841.
Cf. A002260. - Gary W. Adamson, Apr 20 2009
Cf. A052591. - Emeric Deutsch, Jul 18 2009
Cf. A052618, A077611, A199495. - Olivier Gérard, Nov 07 2011

List([0..30], n-> ((-1)^n +2*n +3)*Factorial(n)/4); # G. C. Greubel, May 07 2019

[((-1)^n +2*n +3)*Factorial(n)/4: n in [0..30]]; // G. C. Greubel, May 07 2019

spec := [S,{S=Prod(Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[n!((-1)^n+2n+3)/4,{n,0,30}] (* Harvey P. Dale, Aug 16 2014 *)

PARI
```
a(n)=if(n<0,0,(1+n\2)*n!)
```

a(n)=if(n<0, 0, n!*polcoeff(1/(1-x)/(1-x^2)+x*O(x^n), n))

[((-1)^n +2*n +3)*factorial(n)/4 for n in (0..30)] # G. C. Greubel, May 07 2019

D-finite with recurrence a(n) = a(n-1) + (n^2-1)*a(n-2), with a(1)=1, a(0)=1.
a(n) = ((-1)^n + 2*n + 3)*n!/4.
Let u(1)=1, u(n) = Sum_{k=1..n-1} u(k)*k*(-1)^(k-1) then a(n) = abs(u(n+2)). - Benoit Cloitre, Nov 14 2003
E.g.f.: 1/((1-x)*(1-x^2)).
From Emeric Deutsch, Jul 18 2009: (Start)
a(n) = (n+1)!/2 if n is odd; a(n) = n!(n+2)/2 if n is even.
a(n) = (n+1)! - A052591(n). (End)
E.g.f.: G(0)/(1+x) where G(k) = 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)*(2*k+3)/(x*(2*k+3) + 2*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2012
Sum_{n>=0} 1/a(n) = e - 1/e = 2*sinh(1) (A174548). - Amiram Eldar, Jan 22 2023

A344317 a(n) = n*a(n-1) + n^(1+n mod 2), a(0) = 1.

Original entry on oeis.org

1, 2, 6, 19, 80, 401, 2412, 16885, 135088, 1215793, 12157940, 133737341, 1604848104, 20863025353, 292082354956, 4381235324341, 70099765189472, 1191696008221025, 21450528147978468, 407560034811590893, 8151200696231817880, 171175214620868175481

Offset: 0

Alois P. Heinz, May 14 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A155521, A174548, A344229, A344262.

a:= proc(n) a(n):= n*a(n-1) + n^(1+n mod 2) end: a(0):= 1:
seq(a(n), n=0..23);

E.g.f.: (1+(x+1)*sinh(x))/(1-x).
a(n) = A155521(n-1) + A344262(n) for n > 0.
Lim_{n->infinity} a(n)/n! = 1+2*sinh(1) = 1+e-1/e = 1+A174548. - Amrit Awasthi, May 19 2021

A197222 Numerators of convergents to e - 1/e.

Original entry on oeis.org

2, 5, 7, 40, 47, 275, 597, 872, 3213, 36215, 111858, 148073, 704150, 6485423, 7189573, 49622861, 56812434, 220060163, 716992923, 937053086, 4465205267, 9867463620, 14332668887, 24200132507, 86933066408, 111133198915, 420332663153, 1792463851527

Offset: 1

T. D. Noe, Oct 12 2011

nonn

Cf. A174548 (decimal), A197223 (denominators).

Mathematica
```
Numerator[Convergents[E - 1/E, 40]]
```

A197223 Denominator of convergents to e - 1/e.

Original entry on oeis.org

1, 2, 3, 17, 20, 117, 254, 371, 1367, 15408, 47591, 62999, 299587, 2759282, 3058869, 21112496, 24171365, 93626591, 305051138, 398677729, 1899762054, 4198201837, 6097963891, 10296165728, 36986461075, 47282626803, 178834341484, 762619992739, 941454334223

Offset: 1

T. D. Noe, Oct 12 2011

nonn

Cf. A174548 (decimal), A197222 (numerators).

Mathematica
```
Denominator[Convergents[E - 1/E, 40]]
```

A344418 a(n) = n*a(n-1) + n^(1+n mod 2), a(0) = 0.

Original entry on oeis.org

0, 1, 4, 13, 56, 281, 1692, 11845, 94768, 852913, 8529140, 93820541, 1125846504, 14636004553, 204904063756, 3073560956341, 49176975301472, 836008580125025, 15048154442250468, 285914934402758893, 5718298688055177880, 120084272449158735481, 2641853993881492180604

Offset: 0

Alois P. Heinz, May 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000142, A001113, A073742, A155521, A174548, A344317, A344419.

a:= proc(n) a(n):= n*a(n-1) + n^(1+n mod 2) end: a(0):= 0:
seq(a(n), n=0..23);

E.g.f.: (x+1)*sinh(x)/(1-x).
a(n) = A344317(n) - n! = A344317(n) - A000142(n).
a(n) = A155521(n-1) + A344419(n) for n > 0.
Lim_{n-> infinity} a(n)/n! = 2*sinh(1) = 2*A073742 = e-1/e = A174548. - Amrit Awasthi, May 20 2021

cp's OEIS Frontend

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

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A073742 Decimal expansion of sinh(1).

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A052558 a(n) = n! *((-1)^n + 2*n + 3)/4.

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A344317 a(n) = n*a(n-1) + n^(1+n mod 2), a(0) = 1.

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A197222 Numerators of convergents to e - 1/e.

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A197223 Denominator of convergents to e - 1/e.

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A344418 a(n) = n*a(n-1) + n^(1+n mod 2), a(0) = 0.

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A052558 a(n) = n! ((-1)^n + 2n + 3)/4.