A073742 -id:A073742 - cp's OEIS frontend

A334401 Decimal expansion of sinh(Pi).

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi - e^(-Pi))/2 = 11.5487393572577483779773343153884...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073742, A093580, A156648, A323983, A334402.

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

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

Equals 2 * Product_{k>=1} (4*k^2+4)/(4*k^2-1).

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

A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n(n-1).. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.

Original entry on oeis.org

1, 6, 6, 840, 120, 332640, 5040, 259459200, 362880, 335221286400, 39916800, 647647525324800, 6227020800, 1748648318376960000, 1307674368000, 6288139352883548160000, 355687428096000, 29051203810321992499200000, 121645100408832000, 167683548393178540705382400000

Offset: 1

Amarnath Murthy, Nov 03 2005

			a(3) = 3*2*1 = 6.
a(4) = 4*5*6*7 = 840.

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A113549.

Cf. A019704, A073742, A092042, A092616, A333419, A367563.

n = 1; anfunc[n_] := (If [EvenQ[n], {an = n, Do[an = an*(n + i), {i, n - 1}]}, an = n! ]; an); Table[anfunc[n], {n, 1, 20}] (* Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Mar 10 2006 *)

a(2n-1) = (2n-1)!, a(2n) = (4n-1)!/(2n-1)!.
a(2n-1)*a(2n) = (4n-1)!.
Sum_{n>=1} 1/a(n) = sinh(1) + (sqrt(Pi)/2) * (exp(1/4) * erf(1/2) - exp(-1/4) * erfi(1/2)). - Amiram Eldar, Aug 15 2025

More terms from Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Mar 10 2006

A334367 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!!.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

This constant is transcendental.

			1/(2^1*1!) + 1/(2^3*3!) + 1/(2^5*5!) + ... = 0.52109530549374736162242...

Index entries for transcendental numbers

Cf. A019774, A067626, A073742, A143280, A160327, A201504, A334366.

Mathematica
```
RealDigits[Sinh[1/2], 10, 110] [[1]]
```
PARI
```
sinh(1/2) \\ Michel Marcus, Apr 25 2020
```

Equals sinh(1/2).

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

A078980 Numerators of continued fraction convergents to sinh(1).

Original entry on oeis.org

1, 6, 7, 20, 47, 114, 161, 436, 3213, 16501, 19714, 36215, 55929, 148073, 352075, 6837498, 7189573, 21216644, 28406217, 220060163, 248466380, 468526543, 4465205267, 4933731810, 19266400697, 24200132507, 43466533204, 111133198915

Offset: 0

Benoit Cloitre, Dec 20 2002

Cf. A068139 (continued fraction), A073742 (decimal expansion), A078981 (denominators).

Numerator[Convergents[Sinh[1],30]] (* Harvey P. Dale, May 09 2013 *)

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

Offset changed by Andrew Howroyd, Aug 05 2024

A078981 Denominators of continued fraction convergents to sinh(1).

Original entry on oeis.org

1, 5, 6, 17, 40, 97, 137, 371, 2734, 14041, 16775, 30816, 47591, 125998, 299587, 5818151, 6117738, 18053627, 24171365, 187253182, 211424547, 398677729, 3799524108, 4198201837, 16394129619, 20592331456, 36986461075, 94565253606

Offset: 0

Benoit Cloitre, Dec 20 2002

Cf. A068139 (continued fraction), A073742 (decimal expansion), A078980 (numerators).

Convergents[Sinh[1],30]//Denominator (* Harvey P. Dale, Apr 17 2022 *)

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

Offset changed by Andrew Howroyd, Aug 05 2024

A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

Original entry on oeis.org

1, 60, 5040, 604800, 99792000, 21794572800, 6102480384000, 2134124568576000, 912338253066240000, 468333636574003200000, 284372184127734743040000, 201645730563302817792000000, 165147853331345007771648000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

John Campbell, Applications of Gosper’s nonlocal derangement identity, Maple Transactions, 5, 1, Article 16724, Feb. 2025.

Cf. A002674 (first column of A090438), A091033 (third column), A090438.

Cf. A001620, A049470, A073742, A073743, A099281, A099282, A099284.

a[n_] := (n - 1)*(2*n)!/4!; Array[a, 13, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 3)/8 = (n-1)*(2*n)!/4!
E.g.f.: (-3*hypergeom([1/2, 1], [], 4*x) + hypergeom([1, 3/2], [], 4*x) + 2)/(8*3!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = 60 - 24*Gamma - 24*cosh(1) + 24*CoshIntegral(1) - 24*sinh(1).
Sum_{n>=2} (-1)^n/a(n) = -12 + 24*gamma - 24*cos(1) - 24*CosIntegral(1) + 24*SinIntegral(1). (End)
a(n+1) = Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+2) * binomial(2*n, n-j) (Campbell, Eq. 17). - Peter Bala, Mar 30 2025

A091033 Third column (k=4) of array A090438 ((4,2)-Stirling2).

Original entry on oeis.org

1, 180, 25200, 4233600, 898128000, 239740300800, 79332244992000, 32011868528640000, 15509750302126080000, 8898339094906060800000, 5971815866682429603840000, 4637851802955964809216000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

Cf. A091032 (second column of A090438 divided by 8), A091034 (fourth column divided by 24), A000384, A090438.

Cf. A001620, A049469, A049470, A073742, A099281, A099282, A099283, A099284.

a[n_] := (n-1)*(2*n-3)*(2*n)!/4!; Array[a, 12, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n-3)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 4), n>=2.
a(n) = (n-1)*(2*n-3)*(2*n)!/4! = binomial(2*(n-1), 2)*(2*n)!/4! = A000384(n-1)*(2*n)!/4!, n>=2.
E.g.f.: (6*hypergeom([1/2, 1], [], 4*x) - 4*hypergeom([1, 3/2], [], 4*x) + hypergeom([3/2, 2], [], 4*x) -3)/4! (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = -20 + 24*Gamma - 16*CoshIntegral(1) + 16*sinh(1) + 8*SinhIntegral(1).
Sum_{n>=2} (-1)^n/a(n) = 4 - 24*gamma + 16*cos(1) + 24*CosIntegral(1) - 16*sin(1) + 8*SinIntegral(1). (End)

A091034 Fourth column (k=5) of array A090438 ((4,2)-Stirling2) divided by 24.

Original entry on oeis.org

1, 280, 70560, 19958400, 6659452800, 2644408166400, 1244905998336000, 689322235650048000, 444916954745303040000, 331767548149023866880000, 283424276847308960563200000, 275246422218908346286080000000

Offset: 3

Wolfdieter Lang, Jan 23 2004

Cf. A091033 (third column of A090438), A091035 (fifth column), A090438.

Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

a[n_] := (n - 1)*(n - 2)*(2*n - 3)*(2*n)!/(5!*(3!)^2); Array[a, 12, 3] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2); \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 5)/24, n>=3.
a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2), n>=3.
E.g.f.: (Sum_{p=2..5} (((-1)^(p+1))*binomial(5, p)*hypergeom([(p-1)/2, p/2], [], 4*x)) + 4)/(5!*4!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=3} 1/a(n) = 2010 - 4680*Gamma + 1800*cosh(1) + 4680*CoshIntegral(1) - 2520*sinh(1) - 2880*SinhIntegral(1).
Sum_{n>=3} (-1)^(n+1)/a(n) = -2010 - 3960*gamma + 3240*cos(1) + 3960*CosIntegral(1) - 1800*sin(1) + 2880*SinIntegral(1). (End)

A091035 Fifth column (k=6) of array A090438 ((4,2)-Stirling2).

Original entry on oeis.org

1, 840, 352800, 139708800, 59935075200, 29088489830400, 16183777978368000, 10339833534750720000, 7563588230670151680000, 6303583414831453470720000, 5951909813793488171827200000, 6330667711034891964579840000000

Offset: 3

Wolfdieter Lang, Jan 23 2004

Cf. A091034 (fourth column of A090438 divided by 24), A091036 (sixth column divided by 48), A053134, A090438.

Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

Table[Binomial[2n-2,4] (2n)!/6!,{n,3,20}] (* Harvey P. Dale, Jun 07 2021 *)

a(n) = binomial(2*n-2, 4)*(2*n)!/6!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 6), n>=3.
a(n) = binomial(2*n-2, 4)*(2*n)!/6! = A053134(n-3)*(2*n)!/6!, n>=3.
E.g.f.: (Sum_{p=2..6} (((-1)^p)*binomial(6, p)*hypergeom([(p-1)/2, p/2], [], 4*x)) - 5)/6! (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=3} 1/a(n) = -594 + 1800*Gamma - 1008*cosh(1) - 1800*CoshIntegral(1) + 912*sinh(1) + 1464*SinhIntegral(1).
Sum_{n>=3} (-1)^(n+1)/a(n) = 1554 + 1080*gamma - 1248*cos(1) - 1080*CosIntegral(1) + 240*sin(1) - 1416*SinIntegral(1). (End)

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A334401 Decimal expansion of sinh(Pi).

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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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A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n*(n-1)*.. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.

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A334367 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!!.

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A078980 Numerators of continued fraction convergents to sinh(1).

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Extensions

A078981 Denominators of continued fraction convergents to sinh(1).

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Crossrefs

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Mathematica

PARI

Extensions

A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

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Formula

A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n(n-1).. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.