A073742 -id:A073742 - cp's OEIS frontend

A152885 Number of descents beginning and ending with an odd number in all permutations of {1,2,...,n}.

0, 0, 2, 6, 72, 360, 4320, 30240, 403200, 3628800, 54432000, 598752000, 10059033600, 130767436800, 2440992153600, 36614882304000, 753220435968000, 12804747411456000, 288106816757760000, 5474029518397440000, 133809610449715200000, 2810001819444019200000

Offset: 1

Emeric Deutsch, Jan 19 2009

nonn

a(n) is also number of descents beginning with an odd number and ending with an even number in all permutations of {1,2,...,n}. Example: a(4)=6; indeed for n=4 the only descent to be counted is 32, occurring only in 1324, 1432, 4132, 3214, 3241 and 4321.

			a(6) = 360 because (i) the descent pairs can be chosen in binomial(3,2) = 3 ways, namely (3,1), (5,1), (5,3); (ii) they can be placed in 5 positions, namely (1,2),(2,3),(3,4),(4,5),(5,6); (iii) the remaining 4 entries can be permuted in 4!=24 ways; 3*5*24 = 360.

Cf. A152886, A152887.

Cf. A001620, A073742, A099284.

a := proc (n) if `mod`(n, 2) = 0 then (1/4)*factorial(n)*((1/2)*n-1) else (1/8)*(n-1)*(n+1)*factorial(n-1) end if end proc: seq(a(n), n = 1 .. 20);

a[n_] := (n - 1)! * Binomial[If[OddQ[n], (n + 1)/2, n/2], 2]; Array[a, 25] (* Amiram Eldar, Jan 22 2023 *)

a(2n) = (2n-1)!*binomial(n,2); a(2n+1) = (2n)!*binomial(n+1,2).
D-finite with recurrence (-n+3)*a(n) +(n-1)*a(n-1) +n*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Jul 26 2022
Sum_{n>=3} 1/a(n) = 8*(CoshIntegral(1) - gamma - sinh(1) + 1) = 8*(A099284 - A001620 - A073742 + 1). - Amiram Eldar, Jan 22 2023

A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/1!^2 + 1/3!^2 + 1/5!^2 + 1/7!^2 + ... = 1.027847261597415799692...
Continued fraction: 1 + 1/(36 - 36/(401 - 400/(1765 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n + 1))^2 for n >= 1. - _Peter Bala_, Feb 22 2024

Index entries for sequences related to Bessel functions or polynomials

Cf. A009445, A049469, A070910, A073742, A091681, A197036, A334379.

RealDigits[(BesselI[0, 2] - BesselJ[0, 2])/2, 10, 110] [[1]]

suminf(k=0, 1/((2*k+1)!)^2) \\ Michel Marcus, Apr 26 2020

(besseli(0,2) - besselj(0,2))/2 \\ Michel Marcus, Apr 26 2020

Equals (BesselI(0,2) - BesselJ(0,2))/2.

A334399 Decimal expansion of sinh(e).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 26 2020

			(e^e - e^(-e))/2 = 7.54413710281697582634182004251653274...

Cf. A001113, A073226, A073230, A073742, A085659, A334400.

Mathematica
```
RealDigits[Sinh[E], 10, 110] [[1]]
```

Equals Sum_{k>=0} e^(2*k+1)/(2*k+1)!.

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

A366599 Decimal expansion of arcsinh(e).

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, Oct 14 2023

			1.7253825588523150939450...

Cf. A001113, A073742, A085659, A188640, A334399, A365927.

Mathematica
```
RealDigits[ArcSinh[E], 10, 100] [[1]]
```

asinh(exp(1)) \\ Amiram Eldar, Oct 18 2023

Equals log(A188640). - Amiram Eldar, Oct 18 2023

A052657 Expansion of e.g.f. x^2/((1-x)^2*(1+x)).

Original entry on oeis.org

0, 0, 2, 6, 48, 240, 2160, 15120, 161280, 1451520, 18144000, 199584000, 2874009600, 37362124800, 610248038400, 9153720576000, 167382319104000, 2845499424768000, 57621363351552000, 1094805903679488000, 24329020081766400000, 510909421717094400000, 12364008005553684480000

Offset: 0

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

Stirling transform of -(-1)^n*a(n-1) = [0, 0, 2, -6, 48, -240, ...] is A052841(n-1) = [0, 0, 2, 6, 38, 270, ...]. - Michael Somos, Mar 04 2004

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 604.

Cf. A004526, A052841.

Cf. A001620, A073742, A099284.

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

a[n_] := Floor[n/2] * n!; Array[a, 25, 0] (* Amiram Eldar, Jan 22 2023 *)

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

a(n)=n!*(n\2); \\ Joerg Arndt, Jan 22 2023

a(0)=0, a(1)=0, a(2)=2, n*a(n+2) = (n+2)*a(n+1) + (n^3 + 4*n^2 + 5*n + 2)*a(n).
a(n) = (2*n-1+(-1)^n)*n!/4 = n!*floor(n/2) = n!*A004526(n).
E.g.f.: x^2/((1-x)*(1-x^2)).
Sum_{n>=2} 1/a(n) = 4*CoshIntegral(1) - 4*gamma - 2*sinh(1) + 2 = 4*A099284 - 4*A001620 - 2*A073742 + 2. - Amiram Eldar, Jan 22 2023

A196932 Decimal expansion of sinh(1)*cosh(1).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 23 2011

Also decimal expansion of sinh(2)/2.

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

			sinh(1)*cosh(1) = sinh(2)/2 = 1.8134302039235093838341...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A001113, A073742, A073743.

Sinh(2)/2; // G. C. Greubel, Jun 10 2018

RealDigits[Sinh[2]/2, 10, 100][[1]] (* G. C. Greubel, Jun 10 2018 *)

sinh(2)/2 \\ Charles R Greathouse IV, Apr 21 2016

Equals Product_{k>=1} 1 + 4/(k * Pi)^2. - Amiram Eldar, Jul 16 2020

a(25)-a(72) from John W. Layman, Oct 24 2011

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Jul 10 2021

			0.2288989481961786412366361253722...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A072334, A073742, A073747.

SetDefaultRealField(RealField(135)); 2/(Exp(2)-1)*Exp(2/(1-Exp(2))); // G. C. Greubel, Jun 11 2024

x/.FindRoot[LambertW[-x]-LambertW[-1,-x]==2, {x, 0.1, 0.3}, WorkingPrecision -> 110]
RealDigits[2/(E^2-1)*Exp[2/(1-E^2)], 10, 135][[1]] (* G. C. Greubel, Jun 11 2024 *)

PARI
```
exp(-cotanh(1))/sinh(1)
```

numerical_approx(2/(e^2-1)*exp(2/(1-e^2)), digits=135) # G. C. Greubel, Jun 11 2024

Equals exp(-coth(1))/sinh(1) = exp(-A073747)/A073742.
Equals (coth(1)-1)*exp(1-coth(1)) = (A073747-1)*exp(1-A073747).
Equals (coth(1)+1)/exp(1+coth(1)) = (A073747+1)/exp(1+A073747).
Equals 2/(e^2-1)*exp(2/(1-e^2)) = 2/(A072334^2-1)*exp(2/(1-A072334^2)).

A372338 Decimal expansion of Sum_{k>=0} (k+2) / ((k+1)*(2k)!).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 28 2024

			2.8073217524723591352868580804341...

Cf. A073742, A073743, A372339.

s = Sum[(k + 2)/((k + 1) (2 k)!), {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d]]

Equals -4/e + 2 + 3*cosh(1) - 2*sinh(1).

Equals 2*2F3(1,3; 1/2,2,2; 1/4). - R. J. Mathar, Aug 02 2024

A347199 Decimal expansion of sin(1) * sinh(1).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 22 2021

			0.9888977057628650963821295408926861886421496950...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press (1979), Eq. (1.413.1).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A049469, A073742.

RealDigits[Sin[1] * Sinh[1], 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

sin(1) * sinh(1) \\ Amiram Eldar, Jun 15 2023

Equals Sum_{k>=0} (-1)^k * 2^(2*k+1) / (4*k+2)!.
Equals Product_{k>=1} (1 - 1/(Pi^4*k^4)).
Equals 0F3(;3/4,5/4,3/2; -1/64). - R. J. Mathar, Aug 03 2024

cp's OEIS Frontend

A152885 Number of descents beginning and ending with an odd number in all permutations of {1,2,...,n}.

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Maple

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A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

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PARI

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A334399 Decimal expansion of sinh(e).

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Mathematica

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

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Maple

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A366599 Decimal expansion of arcsinh(e).

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Mathematica

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A052657 Expansion of e.g.f. x^2/((1-x)^2*(1+x)).

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Maple

Mathematica

PARI

PARI

Formula

A196932 Decimal expansion of sinh(1)*cosh(1).

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Magma

Mathematica

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Extensions

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.