A073742 -id:A073742 - cp's OEIS frontend

A174548 Decimal expansion of e - 1/e.

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

Offset: 1

Paul Curtz, Mar 22 2010

			2.3504023872876029137647637...

Index entries for transcendental numbers

Cf. A001113 (e), A068985 (1/e), and A137204 (e + 1/e), A073742 (sinh(1)).

evalf(exp(1)-exp(-1)) ; # R. J. Mathar, Oct 14 2011

Mathematica
```
RealDigits[E - 1/E, 10, 111][[1]]
```

exp(1) - exp(-1) \\ Michel Marcus, May 05 2019

Equals 2 * sinh(1) = 2 * A073742. - Amiram Eldar, Nov 25 2020

Edited and extended by Robert G. Wilson v, Apr 25 2010

A068139 Continued fraction expansion for sinh(1).

Original entry on oeis.org

1, 5, 1, 2, 2, 2, 1, 2, 7, 5, 1, 1, 1, 2, 2, 19, 1, 2, 1, 7, 1, 1, 9, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 4, 5, 3, 5, 1, 3, 1, 1, 1, 2, 7, 1, 9, 1, 1, 2, 1, 21, 1, 18, 1, 2, 734, 1, 2, 1, 84, 2, 2, 1, 1, 2, 10, 1, 3, 7, 1, 16, 1, 2, 4, 56, 1, 13, 16, 208

Offset: 0

Benoit Cloitre, Mar 13 2002

If an extra zero is added to the beginning of this sequence, continued fraction for csch(1) = 1/sinh(1). - Rick L. Shepherd, Aug 07 2002

Robert Israel, Table of n, a(n) for n = 0..9999

Cf. A068118, A073742 (decimal expansion), A073745 (decimal expansion of csch(1)).

Cf. A078980, A078981 (convergents).

numtheory:-cfrac(sinh(1),100,quotients)[1..-2]; # Robert Israel, Nov 14 2020

ContinuedFraction[Sinh@ 1, 82] (* Michael De Vlieger, Sep 04 2016 *)

contfrac(sinh(1)) \\ Michel Marcus, Sep 04 2016

Offset changed by Andrew Howroyd, Aug 05 2024

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).

Original entry on oeis.org

0, 1, 2, 2, 8, 24, 144, 720, 5760, 40320, 403200, 3628800, 43545600, 479001600, 6706022400, 87178291200, 1394852659200, 20922789888000, 376610217984000, 6402373705728000, 128047474114560000, 2432902008176640000, 53523844179886080000, 1124000727777607680000

Offset: 0

Paul Barry, Sep 14 2004

G. C. Greubel, Table of n, a(n) for n = 0..450

From Johannes W. Meijer, Nov 12 2009: (Start)
Cf. A109613 (odd numbers repeated).
Equals the first left hand column of A167552.
Equals the first right hand column of A167556.
A098557(n)*A064455(n) equals the second right hand column of A167556(n).
(End)
Cf. A052558, A073742.

[0,1] cat [Factorial(n-1) + Factorial(n-2)*(1+(-1)^n)/2: n in [2..30]]; // G. C. Greubel, Jan 17 2018

Join[{0,1}, Table[(n-1)! + (n-2)!*(1+(-1)^n)/2, {n,2,30}]] (* or *) With[{nmax = 50}, CoefficientList[Series[(1/2)*(1 + x)*Log[(1 + x)/(1 - x)], {x,0,nmax}], x]*Range[0,nmax]!] (* G. C. Greubel, Jan 17 2018 *)

for(n=0, 30, print1(if(n==0,0, if(n==1, 1, (n-1)! + (n-2)!*(1 + (-1)^n)/2)), ", ")) \\ G. C. Greubel, Jan 17 2018

a(n+1) = n! + (n-1)! * (1-(-1)^n)/2.
a(n+2) = 2*A052558(n).
conjecture: -a(n) +a(n-1) +(n-1)*(n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
G.f.: 1-G(0), where G(k)= 1 + x*(2*k-1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
Sum_{n>=1} 1/a(n) = sinh(1) + 1 = A073742 + 1. - Amiram Eldar, Jan 22 2023

A052618 Expansion of e.g.f. 1/((1-x)^2*(1-x^2)).

Original entry on oeis.org

1, 2, 8, 36, 216, 1440, 11520, 100800, 1008000, 10886400, 130636800, 1676505600, 23471078400, 348713164800, 5579410636800, 94152554496000, 1694745980928000, 32011868528640000, 640237370572800000, 13380961044971520000, 294381142989373440000

Offset: 0

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

Permanent of the n X n (0, 1)-matrix with (i, j)-th entry equal to 0 iff (i=1, j=n), (i=2, j=1), (i=3, j=n), (i=4, j=1), ... - Simone Severini, Oct 17 2004
a(n) is the number of runs of odd entries in all permutations of {1,2,...,n+1}. Example: a(2)=8 because in the permutations 123, 132, 213, 231, 312 and 321 we have a total of 2+1+1+1+1+2 runs of odd entries. - Emeric Deutsch, Dec 14 2008
a(n) is the number of permutations of [n+2] whose first place is even and last place is odd (or any equivalent definition with two separate places in a permutation). - Olivier Gérard, Nov 07 2011

			The a(2) = 8 permutations of [4] starting with an even number and ending with an odd number are: 2143, 2341, 2413, 2431, 4123, 4213, 4231, 4321.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 563.

Cf. A002620, A152666.

Cf. A001620, A073742, A099284.

spec := [S,{S=Prod(Sequence(Z),Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a := proc (n) options operator, arrow: factorial(n)*floor((1/2)*n+1)*ceil((1/2)*n+1) end proc; seq(a(n), n = 0 .. 20); # Emeric Deutsch, Dec 14 2008

With[{nn=20},CoefficientList[Series[1/((1-x)^2*(1-x^2)),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 01 2019 *)

E.g.f.: -1/(-1+x)^2/(-1+x^2).
Recurrence: {a(0)=1, a(1)=2, (-n^2-5*n-4)*a(n)+a(n+2)-2*a(n+1)=0.}.
a(n) = (1/8*(-1)^(-n)+1/4*n^2+n+7/8)*n! for n>0.
From Emeric Deutsch, Dec 14 2008: (Start)
a(n) = n!*floor((n+2)/2)*ceiling((n+2)/2).
a(n) = Sum_{k>=1} (k*A152666(n+1,k)). (End)
a(n) = n!*A002620(n+2). - R. J. Mathar, Nov 27 2011
Sum_{n>=0} 1/a(n) = 4*(sinh(1) + gamma - CoshIntegral(1)) - 2 = 4*(A073742 + A001620 - A099284) - 2. - Amiram Eldar, Jan 22 2023

A077611 Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,...,n}.

Original entry on oeis.org

0, 0, 4, 12, 144, 720, 8640, 60480, 806400, 7257600, 108864000, 1197504000, 20118067200, 261534873600, 4881984307200, 73229764608000, 1506440871936000, 25609494822912000, 576213633515520000, 10948059036794880000, 267619220899430400000, 5620003638888038400000

Offset: 1

Leroy Quet, Frank Ruskey, and Dean Hickerson, Nov 11 2002

nonn

a(n) is also the number of permutations of [n+1] starting and ending with an even number. - Olivier Gérard, Nov 07 2011

			For n=4, the a(4) = 12 permutations of degree 5 starting and ending with an even number are 21354, 21534, 23154, 23514, 25134, 25314, 41352, 41532, 43152, 43512, 45132, 45312.

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Cf. A052618, A077612, A077613.

Cf. A001620, A073742, A099284.

[Factorial(n-1)*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8 : n in [1..30]]; // Vincenzo Librandi, Nov 16 2011

Table[Ceiling[n/2] Ceiling[n/2 - 1] (n - 1)!, {n, 22}] (* Michael De Vlieger, Aug 20 2017 *)

a(n) = ceiling(n/2)*ceiling(n/2-1)*(n-1)!. Proof: There are ceiling(n/2) * ceiling(n/2-1) pairs (r, s) with r and s odd and distinct. For each pair, there are n-1 places it can occur in a permutation and (n-2)! possible arrangements of the other numbers.
a(n) = (n-1)!*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8. - Bruno Berselli, Nov 07 2011
Sum_{n>=3} 1/a(n) = 4*(CoshIntegral(1) - gamma - sinh(1) + 1) = 4*(A099284 - A001620 - A073742 + 1). - Amiram Eldar, Jan 22 2023

A109234 a(n) = floor(n*sinh(1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77

Offset: 1

Reinhard Zumkeller, Jun 23 2005

nonn

Beatty sequence for sinh(1) = (e-1/e)/2 = 1.17520... = A073742; complement of A109235.

Eric Weisstein's World of Mathematics, Hyperbolic Sine
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A109236, A109231, A109237.

A354211 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.

Original entry on oeis.org

1, 7, 47, 5923, 426457, 15636757, 7318002277, 1536780478171, 603180793741, 142957467201379447, 60042136224579367741, 10127106976545720025649, 18228792557782296046168201, 12796612375563171824410077103, 3463616416319098507140327535879, 1380498543075754976417359117871773

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 7/6, 47/40, 5923/5040, 426457/362880, 15636757/13305600, 7318002277/6227020800, ...

Cf. A009445, A053557, A061354, A073742, A103816, A120265, A143382, A289381, A354331 (denominators), A354332, A354334.

Table[Sum[1/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sinh[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator

a(n) = numerator(sum(k=0, n, 1/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354211(n): return sum(Fraction(1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Numerators of coefficients in expansion of sinh(sqrt(x)) / (sqrt(x) * (1 - x)).

A354331 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k+1)!.

Original entry on oeis.org

1, 6, 40, 5040, 362880, 13305600, 6227020800, 1307674368000, 513257472000, 121645100408832000, 51090942171709440000, 8617338912961658880000, 15511210043330985984000000, 10888869450418352160768000000, 2947253997913233984847872000000, 1174691236311131831103651840000000

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 7/6, 47/40, 5923/5040, 426457/362880, 15636757/13305600, 7318002277/6227020800, ...

Cf. A009445, A053556, A061355, A073742, A143383, A289488, A354211 (numerators), A354333, A354335.

Table[Sum[1/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Sinh[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Denominator

a(n) = denominator(sum(k=0, n, 1/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354331(n): return sum(Fraction(1,factorial(2*k+1)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022

Denominators of coefficients in expansion of sinh(sqrt(x)) / (sqrt(x) * (1 - x)).

A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

			1/1! + 1/5! + 1/9! + ... = 1.008336089225848981767442...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 36.

Cf. A001113, A049469, A073742, A143820, A332890, A334364, A334365.

RealDigits[(Sin[1] + Sinh[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+1)!) \\ Michel Marcus, Apr 25 2020

Equals (sin(1) + sinh(1))/2.

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/3! + 1/7! + 1/11! + ... = 0.1668651044179524751...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.
Index entries for transcendental numbers.

Cf. A001113, A049469, A073742, A332890, A334363, A334364.

RealDigits[(Sinh[1] - Sin[1])/2, 10, 111] [[1]]

suminf(k=0, 1/(4*k+3)!) \\ Michel Marcus, Apr 25 2020

Equals (sinh(1) - sin(1))/2.

cp's OEIS Frontend

A174548 Decimal expansion of e - 1/e.

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A068139 Continued fraction expansion for sinh(1).

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A098557 Expansion of e.g.f. (1/2)*(1+x)*log((1+x)/(1-x)).

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A052618 Expansion of e.g.f. 1/((1-x)^2*(1-x^2)).

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A077611 Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,...,n}.

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A109234 a(n) = floor(n*sinh(1)).

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A354211 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.

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A354331 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k+1)!.

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).