A073743 -id:A073743 - cp's OEIS frontend

A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

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

Offset: 1

Ilya Gutkovskiy, Feb 21 2016

From Vaclav Kotesovec, Feb 24 2016: (Start)
Sum_{k>=0} 1/k! = A001113 = exp(1).
Sum_{k>=0} 1/(2k)! = A073743 = cosh(1).
Sum_{k>=0} 1/(3k)! = A143819 = (2*cos(sqrt(3)/2)*exp(-1/2) + exp(1))/3.
Sum_{k>=0} 1/(4k)! = (cos(1) + cosh(1))/2 = 1.0416914703416917479394211141...
(End)
For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity. - Bernard Schott, Mar 02 2020
Continued fraction: 1 + 1/(120 - 120/(30241 - 30240/(360361 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (5*n)*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

			1 + 1/5! + 1/10! + 1/15! + ... = 1.008333608907290289976453667354838786...

Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A100734.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), this sequence (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

evalf((exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5, 120); # Vaclav Kotesovec, Feb 24 2016

RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, 1/3125], 10, 120][[1]]

suminf(k=0, 1/(5*k)!) \\ Michel Marcus, Feb 21 2016

Equals Sum_{k>=0} 1/A100734(k).
Equals (exp(1) + exp(-(-1)^(1/5)) + exp((-1)^(2/5)) + exp(-(-1)^(3/5)) + exp((-1)^(4/5)))/5.
Equals (exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5. - Vaclav Kotesovec, Feb 24 2016
Sum_{k>=0} (-1)^k / (5*k)! = (exp(-1) + 2*cos(5^(1/4)/(2*sqrt(phi))) * exp(phi/2) + 2*cos(5^(1/4)*sqrt(phi)/2) / exp(1/(2*phi)))/5 = 0.99166694223909419..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2020

A275651 a(n) = (2n)!Sum_{k = 0..n} (-1)^k/(2*k)!.

Original entry on oeis.org

1, 1, 13, 389, 21785, 1960649, 258805669, 47102631757, 11304631621681, 3459217276234385, 1314502564969066301, 607300185015708631061, 335229702128671164345673

Offset: 0

Peter Bala, Sep 02 2016

Compare with the derangement numbers A000166(n) := n!*sum_{k = 0..n} (-1)^k/k! and also A074790.

Seiichi Manyama, Table of n, a(n) for n = 0..225

Cf. A000166, A049470 (cos(1)), A051396, A073743, A074790.

A275651 := proc(n) option remember; if (n = 0) then 1 else 2*n*(2*n - 1)*A275651(n-1)+(-1)^n end if; end proc:
seq(A275651(n), n = 0..20);

Table[(2 n)!*Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 12}] (* Michael De Vlieger, Sep 04 2016 *)

a(n) ~ (2*n)!*cos(1).
E.g.f. for the aerated sequence: cos(x)/(1 - x^2) = 1 + x^2/2! + 13*x^4/4! + 389*x^6/6! + ....
Recurrence equations:
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 - 2)*(2*n - 3)*a(n - 2) with a(0) = 1, a(1) = 1.
The latter recurrence is also satisfied by the sequence b(n) := (2*n)! with b(0) = 1, b(1) = 2. This leads to the continued fraction representation a(n) = (2*n )!*( 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/(4*n^2 - 2*n - 1) )))) ) for n >= 3. Taking the limit gives the continued fraction representation cos(1) = A049470 = 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/((4*n^2 - 2*n - 1) + ... ))))). Cf. A073743.

A332892 Decimal expansion of Sum_{k>=0} 1/(6*k)!.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 02 2020

For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity.

			1.001388890976564743867770084409737409278656173555781142...

Serge Francinou, Hervé Gianella, Serge Nicolas, Exercices de Mathématiques, Oraux X-ENS, Analyse 2, problème 3.10, p. 182, Cassini, Paris, 2004

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), this sequence (Sum 1/(6k)!), A346441.

Maple
```
evalf(sum(1/(6*n)!,n=0..infinity),150);
```

RealDigits[(1/3)*(Cosh[1] + 2*Cosh[1/2]*Cos[Sqrt[3]/2]), 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

sumpos(k=0, 1/(6*k)!) \\ Michel Marcus, Mar 02 2020

Equals (1/3) * (cosh(1) + 2*cosh(1/2)*cos((sqrt(3))/2)).
Sum_{k>=0} (-1)^k / (6*k)! = (cos(1) + 2*cos(1/2)*cosh(sqrt(3)/2))/3 = 0.9986111131987866537... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(720 - 720/(665281 - 665280/(13366081 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (6*n)*(6*n - 1)*(6*n - 2)*(6*n - 3)*(6*n - 4)*(6*n - 5) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

A334364 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/2! + 1/6! + 1/10! + ... = 0.5013891644735520305...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 457.

Cf. A001113, A049470, A073743, A143821, A332890, A334363, A334365.

RealDigits[(Cosh[1] - Cos[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+2)!) \\ Michel Marcus, Apr 25 2020

Equals (cosh(1) - cos(1))/2.

A078982 Denominators of continued fraction convergents to cosh(1).

Original entry on oeis.org

1, 1, 2, 11, 35, 116, 267, 383, 8310, 8693, 17003, 25696, 119787, 265270, 385057, 18748006, 1331493483, 9339202387, 10670695870, 20009898257, 190759780183, 210769678440, 612299137063, 823068815503, 9666056107596, 10489124923099

Offset: 0

Benoit Cloitre, Dec 20 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..199

Cf. A068118 (continued fraction), A073743 (decimal expansion), A078983 (numerators).

Denominator[Convergents[Cosh[1],30]] (* Harvey P. Dale, Jul 22 2014 *)

a(n)=component(component(contfracpnqn(contfrac(cosh(1),n+1)),1),2)

Offset changed by Andrew Howroyd, Aug 05 2024

A078983 Numerators of continued fraction convergents to cosh(1).

Original entry on oeis.org

1, 2, 3, 17, 54, 179, 412, 591, 12823, 13414, 26237, 39651, 184841, 409333, 594174, 28929685, 2054601809, 14411142348, 16465744157, 30876886505, 294357722702, 325234609207, 944826941116, 1270061550323, 14915503994669, 16185565544992

Offset: 0

Benoit Cloitre, Dec 20 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..199

Cf. A068118 (continued fraction), A073743 (decimal expansion), A078982 (denominators).

Numerator[Convergents[Cosh[1],30]] (* Harvey P. Dale, Feb 02 2012 *)

a(n)=component(component(contfracpnqn(contfrac(cosh(1),n+1)),1),1)

Offset changed by Andrew Howroyd, Aug 05 2024

A107991 Complexity (number of maximal spanning trees) in an unoriented simple graph with nodes {1,2,...,n} and edges {i,j} if i + j > n.

Original entry on oeis.org

1, 1, 1, 3, 8, 40, 180, 1260, 8064, 72576, 604800, 6652800, 68428800, 889574400, 10897286400, 163459296000, 2324754432000, 39520825344000, 640237370572800, 12164510040883200, 221172909834240000, 4644631106519040000, 93666727314800640000

Offset: 1

Roland Bacher, Jun 13 2005

nonn

Proof of the formula: check that the associated combinatorial Laplacian has eigenvalues {0,..n-1}\ {floor((n+1)/2)} by exhibiting a basis of eigenvectors (which are very simple).

			a(1)=a(2)=a(3)=1 because the corresponding graphs are trees.
a(4)=3 because the corresponding graph is a triangle with one of its vertices adjacent to a fourth vertex.

N. Biggs, Algebraic Graph Theory, Cambridge University Press (1974).

Vincenzo Librandi, Table of n, a(n) for n = 1..450
Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, Bootstrapping cascaded random matrix models: correlations in permutations of matrix products, arXiv:2405.02541 [math-ph], 2024. See p. 7.
Pierre-Alain Sallard, Coefficients of repeated integrals of hyperbolic cosine.

Cf. A001113, A073742, A073743.

List([1..20],n->Factorial(n-1)/Int((n+1)/2)); # Muniru A Asiru, Dec 15 2018

[Factorial(n-1)/Floor((n+1)/2): n in [1..25]]; // Vincenzo Librandi, Dec 15 2018

Maple
```
a:=n->(n-1)!/floor((n+1)/2);
```

Function[x, 1/x] /@
CoefficientList[Series[3*Exp[x]/4 + 1/4*Exp[-x] + x/2*Exp[x], {x, 0, 10}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)
Table[(n - 1)! / Floor[(n + 1) / 2], {n, 1, 30}] (* Vincenzo Librandi, Dec 15 2018 *)

A107991(n)=(n-1)!/round(n/2) \\ M. F. Hasler, Apr 21 2015

[factorial(n-1)/floor((n+1)/2) for n in range(1,24)] # Stefano Spezia, May 10 2024

a(n) = (n-1)!/floor((n+1)/2).
a(n+1) = n!/floor(n/2 + 1). - M. F. Hasler, Apr 21 2015
1/a(n+1) is the coefficient of the power series of 3*exp(x)/4 + 1/4*exp(-x) + x/2*exp(x) ; this function is the sum of f_n(x) where f_0(x)=cosh(x) and f_{n+1} is the primitive of f_n. - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (e + sinh(1))/2 + cosh(1). - Amiram Eldar, Aug 15 2025

A334379 Decimal expansion of Sum_{k>=0} 1/((2*k)!)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/0!^2 + 1/2!^2 + 1/4!^2 + 1/6!^2 + ... = 1.25173804073865146774451594773...

Index entries for sequences related to Bessel functions or polynomials

Cf. A010050, A049470, A070910, A073743, A091681, A197036, A334378.

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

suminf(k=0, 1/((2*k)!)^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.

Continued fraction: 1 + 1/(4 - 4/(145 - 144/(901 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n - 1))^2. - Peter Bala, Feb 22 2024

A334402 Decimal expansion of cosh(Pi).

Original entry on oeis.org

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi + e^(-Pi))/2 = 11.5919532755215206277517520525601...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073743, A093580, A175314, A308715, A323982, A334401.

Mathematica
```
RealDigits[Cosh[Pi], 10, 110] [[1]]
```

Equals Sum_{k>=0} Pi^(2*k)/(2*k)!.
Equals Product_{k>=0} (1 + 4/(2*k+1)^2).
Equals Product_{k>=1} (k^2 + 4)/(k^2 + 1). - Amiram Eldar, Aug 09 2020

A344419 a(n) = n*a(n-1) + n^(n mod 2), a(0) = 0.

Original entry on oeis.org

0, 1, 3, 12, 49, 250, 1501, 10514, 84113, 757026, 7570261, 83272882, 999274585, 12990569618, 181867974653, 2728019619810, 43648313916961, 742021336588354, 13356384058590373, 253771297113217106, 5075425942264342121, 106583944787551184562, 2344846785326126060365

Offset: 0

Alois P. Heinz, May 17 2021

nonn

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

Cf. A000142, A001113, A073743, A155521, A137204, A344262, A344418.

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

E.g.f.: ((x+1)*cosh(x)-1)/(1-x).
a(n) = A344262(n) - n! = A344262(n) - A000142(n).
a(n) = A344418(n) - A155521(n-1) for n > 0.
Lim_{n->infinity} a(n)/n! = 2*cosh(1)-1 = 2*A073743-1 = e+1/e-1 = A137204-1. - Amrit Awasthi, May 20 2021

cp's OEIS Frontend

A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

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A275651 a(n) = (2*n)!*Sum_{k = 0..n} (-1)^k/(2*k)!.

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A332892 Decimal expansion of Sum_{k>=0} 1/(6*k)!.

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A334364 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!.

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A078982 Denominators of continued fraction convergents to cosh(1).

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A078983 Numerators of continued fraction convergents to cosh(1).

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Mathematica

PARI

Extensions

A107991 Complexity (number of maximal spanning trees) in an unoriented simple graph with nodes {1,2,...,n} and edges {i,j} if i + j > n.

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A275651 a(n) = (2n)!Sum_{k = 0..n} (-1)^k/(2*k)!.