A092514 -id:A092514 - cp's OEIS frontend

A052562 a(n) = 5^n * n!.

1, 5, 50, 750, 15000, 375000, 11250000, 393750000, 15750000000, 708750000000, 35437500000000, 1949062500000000, 116943750000000000, 7601343750000000000, 532094062500000000000, 39907054687500000000000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

A simple regular expression in a labeled universe.

For n >= 1 a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_5)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 504
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000142, A008548, A008546, A034325, A000165, A047055, A047056, A051150, A092514, A092618.

[5^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011

spec := [S,{S=Sequence(Union(Z,Z,Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combstruct):A:=[N,{N=Cycle(Union(Z$5))},labeled]: seq(count(A,size=n)/5,n=1..16); # Zerinvary Lajos, Dec 05 2007

Table[5^n*n!, {n, 0, 20}] (* Wesley Ivan Hurt, Sep 28 2013 *)

{a(n) = 5^n*n!}; \\ G. C. Greubel, May 05 2019

[5^n*factorial(n) for n in (0..20)] # G. C. Greubel, May 05 2019

a(n) = A051150(n+1, 0) (first column of triangle).
E.g.f.: 1/(1-5*x).
a(n) = 5*n*a(n-1) with a(0)=1.
G.f.: 1/(1-5*x/(1-5*x/(1-10*x/(1-10*x/(1-15*x/(1-15*x/(1-20*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
G.f.: 1/Q(0), where Q(k) = 1 - 5*x*(2*k+1) - 25*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
a(n) = n!*A000351(n). - R. J. Mathar, Aug 21 2014
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/5) (A092514).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/5) (A092618). (End)

Name changed by Arkadiusz Wesolowski, Oct 04 2011

A195267 a(n) = 2n^(2n)(1+ sum((1+k^(n+2)/n+k/n)/(n^kk!), k=1..infinity)) / exp(1/n).

Original entry on oeis.org

14, 89, 3104, 265832, 42711550, 11454711984, 4840928618302, 3108267243405312, 2924919606477510504, 3870562086323220000000, 6908112063470228210507358, 16038451894952529045493383168, 47076979725683125083464370176990, 170928617514191160703263296201650176

Offset: 1

Alexander R. Povolotsky, Sep 14 2011

nonn

			n=1: 2*(1+ sum((1+k^3+k)/k!, k=1..infinity))/exp(1) = 14.

Cf. A001113, A092516, A092514, A092041.

A267318 Continued fraction expansion of e^(1/5).

Original entry on oeis.org

1, 4, 1, 1, 14, 1, 1, 24, 1, 1, 34, 1, 1, 44, 1, 1, 54, 1, 1, 64, 1, 1, 74, 1, 1, 84, 1, 1, 94, 1, 1, 104, 1, 1, 114, 1, 1, 124, 1, 1, 134, 1, 1, 144, 1, 1, 154, 1, 1, 164, 1, 1, 174, 1, 1, 184, 1, 1, 194, 1, 1, 204, 1, 1, 214, 1, 1, 224, 1, 1, 234, 1, 1, 244, 1, 1, 254, 1, 1, 264, 1, 1

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

e^(1/5) is a transcendental number.

In general, the ordinary generating function for the continued fraction expansion of e^(1/k), with k = 1, 2, 3..., is (1 + (k - 1)*x + x^2 - (k + 1)*x^3 + 7*x^4 - x^5)/(1 - x^3)^2.

			e^(1/5) = 1 + 1/(4 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + 1/...))))).

Eric Weisstein's World of Mathematics, e Continued Fraction
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A092514.

Cf. continued fraction expansion of e^(1/k): A003417 (k=1), A058281 (k=2), A078689 (k=3), A078688 (k=4), this sequence (k=5).

[1+(3+10*Floor(n/3))*(1-(n-1)^2 mod 3): n in [0..90]]; // Bruno Berselli, Feb 04 2016

ContinuedFraction[Exp[1/5], 82]
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 4, 1, 1, 14, 1}, 82]
CoefficientList[Series[(1 + 4 x + x^2 - x^3 + 6 x^4 - x^5) / (x^3 - 1)^2, {x, 0, 70}], x] (* Vincenzo Librandi, Jan 13 2016 *)
Table[1 + (3 + 10 Floor[n/3]) (1 - Mod[(n - 1)^2, 3]), {n, 0, 90}] (* Bruno Berselli, Feb 04 2016 *)

G.f.: (1 + 4*x + x^2 - x^3 + 6*x^4 - x^5)/(1 - x^3)^2.

a(n) = 1 + (3 + 10*floor(n/3))*(1 - (n-1)^2 mod 3). [Bruno Berselli, Feb 04 2016]

Edited by Bruno Berselli, Feb 04 2016

cp's OEIS Frontend

A052562 a(n) = 5^n * n!.

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A195267 a(n) = 2n^(2n)(1+ sum((1+k^(n+2)/n+k/n)/(n^kk!), k=1..infinity)) / exp(1/n).

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A267318 Continued fraction expansion of e^(1/5).

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cp's OEIS Frontend

A052562 a(n) = 5^n * n!.

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Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A195267 a(n) = 2*n^(2*n)*(1+ sum((1+k^(n+2)/n+k/n)/(n^k*k!), k=1..infinity)) / exp(1/n).

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A267318 Continued fraction expansion of e^(1/5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A195267 a(n) = 2n^(2n)(1+ sum((1+k^(n+2)/n+k/n)/(n^kk!), k=1..infinity)) / exp(1/n).