cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A000165 Double factorial of even numbers: (2n)!! = 2^n*n!.

Original entry on oeis.org

1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000, 1678343852714360832000, 63777066403145711616000
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is also the size of the automorphism group of the graph (edge graph) of the n-dimensional hypercube and also of the geometric automorphism group of the hypercube (the two groups are isomorphic). This group is an extension of an elementary Abelian group (C_2)^n by S_n. (C_2 is the cyclic group with two elements and S_n is the symmetric group.) - Avi Peretz (njk(AT)netvision.net.il), Feb 21 2001
Then a(n) appears in the power series: sqrt(1+sin(y)) = Sum_{n>=0} (-1)^floor(n/2)*y^(n)/a(n) and sqrt((1+cos(y))/2) = Sum_{n>=0} (-1)^n*y^(2n)/a(2n). - Benoit Cloitre, Feb 02 2002
Appears to be the BinomialMean transform of A001907. See A075271. - John W. Layman, Sep 28 2002
Number of n X n monomial matrices with entries 0, +-1.
Also number of linear signed orders.
Define a "downgrade" to be the permutation d which places the items of a permutation p in descending order. This note concerns those permutations that are equal to their double-downgrades. The number of permutations of order 2n having this property are equinumerous with those of order 2n+1. a(n) = number of double-downgrading permutations of order 2n and 2n+1. - Eugene McDonnell (eemcd(AT)mac.com), Oct 27 2003
a(n) = (Integral_{x=0..Pi/2} cos(x)^(2*n+1) dx) where the denominators are b(n) = (2*n)!/(n!*2^n). - Al Hakanson (hawkuu(AT)excite.com), Mar 02 2004
1 + (1/2)x - (1/8)x^2 - (1/48)x^3 + (1/384)x^4 + ... = sqrt(1+sin(x)).
a(n)*(-1)^n = coefficient of the leading term of the (n+1)-th derivative of arctan(x), see Hildebrand link. - Reinhard Zumkeller, Jan 14 2006
a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is j for iDavid Callan, Sep 25 2006
a(n) is the number of increasing plane trees with n+1 edges. (In a plane tree, each subtree of the root is an ordered tree but the subtrees of the root may be cyclically rotated.) Increasing means the vertices are labeled 0,1,2,...,n+1 and each child has a greater label than its parent. Cf. A001147 for increasing ordered trees, A000142 for increasing unordered trees and A000111 for increasing 0-1-2 trees. - David Callan, Dec 22 2006
Hamed Hatami and Pooya Hatami prove that this is an upper bound on the cardinality of any minimal dominating set in C_{2n+1}^n, the Cartesian product of n copies of the cycle of size 2n+1, where 2n+1 is a prime. - Jonathan Vos Post, Jan 03 2007
This sequence and (1,-2,0,0,0,0,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Oct 29 2007
a(n) = number of permutations of the multiset {1,1,2,2,...,n,n,n+1,n+1} such that between the two occurrences of i, there is exactly one entry >i, for i=1,2,...,n. Example: a(2) = 8 counts 121323, 131232, 213123, 231213, 232131, 312132, 321312, 323121. Proof: There is always exactly one entry between the two 1s (when n>=1). Given a permutation p in A(n) (counted by a(n)), record the position i of the first 1, then delete both 1s and subtract 1 from every entry to get a permutation q in A(n-1). The mapping p -> (i,q) is a bijection from A(n) to the Cartesian product [1,2n] X A(n-1). - David Callan, Nov 29 2007
Row sums of A028338. - Paul Barry, Feb 07 2009
a(n) is the number of ways to seat n married couples in a row so that everyone is next to their spouse. Compare A007060. - Geoffrey Critzer, Mar 29 2009
From Gary W. Adamson, Apr 21 2009: (Start)
Equals (-1)^n * (1, 1, 2, 8, 48, ...) dot (1, -3, 5, -7, 9, ...).
Example: a(4) = 384 = (1, 1, 2, 8, 48) dot (1, -3, 5, -7, 9) = (1, -3, 10, -56, 432). (End)
exp(x/2) = Sum_{n>=0} x^n/a(n). - Jaume Oliver Lafont, Sep 07 2009
Assuming n starts at 0, a(n) appears to be the number of Gray codes on n bits. It certainly is the number of Gray codes on n bits isomorphic to the canonical one. Proof: There are 2^n different starting positions for each code. Also, each code has a particular pattern of bit positions that are flipped (for instance, 1 2 1 3 1 2 1 for n=3), and these bit position patterns can be permuted in n! ways. - D. J. Schreffler (ds1404(AT)txstate.edu), Jul 18 2010
E.g.f. of 0,1,2,8,... is x/(1-2x/(2-2x/(3-8x/(4-8x/(5-18x/(6-18x/(7-... (continued fraction). - Paul Barry, Jan 17 2011
Number of increasing 2-colored trees with choice of two colors for each edge. In general, if we replace 2 with k we get the number of increasing k-colored trees. For example, for k=3 we get the triple factorial numbers. - Wenjin Woan, May 31 2011
a(n) = row sums of triangle A193229. - Gary W. Adamson, Jul 18 2011
Also the number of permutations of 2n (or of 2n+1) that are equal to their reverse-complements. (See the Egge reference.) Note that the double-downgrade described in the preceding comment (McDonnell) is equivalent to the reverse-complement. - Justin M. Troyka, Aug 11 2011
The e.g.f. can be used to form a generator, [1/(1-2x)] d/dx, for A000108, so a(n) can be applied to A145271 to generate the Catalan numbers. - Tom Copeland, Oct 01 2011
The e.g.f. of 1/a(n) is BesselI(0,sqrt(2*x)). See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 09 2012
a(n) = order of the largest imprimitive group of degree 2n with n systems of imprimitivity (see [Miller], p. 203). - L. Edson Jeffery, Feb 05 2012
Row sums of triangle A208057. - Gary W. Adamson, Feb 22 2012
a(n) is the number of ways to designate a subset of elements in each n-permutation. a(n) = A000142(n) + A001563(n) + A001804(n) + A001805(n) + A001806(n) + A001807(n) + A035038(n) * n!. - Geoffrey Critzer, Nov 08 2012
For n>1, a(n) is the order of the Coxeter groups (also called Weyl groups) of types B_n and C_n. - Tom Edgar, Nov 05 2013
For m>0, k*a(m-1) is the m-th cumulant of the chi-squared probability distribution for k degrees of freedom. - Stanislav Sykora, Jun 27 2014
a(n) with 0 prepended is the binomial transform of A120765. - Vladimir Reshetnikov, Oct 28 2015
Exponential self-convolution of A001147. - Vladimir Reshetnikov, Oct 08 2016
Also the order of the automorphism group of the n-ladder rung graph. - Eric W. Weisstein, Jul 22 2017
a(n) is the order of the group O_n(Z) = {A in M_n(Z): A*A^T = I_n}, the group of n X n orthogonal matrices over the integers. - Jianing Song, Mar 29 2021
a(n) is the number of ways to tile a (3n,3n)-benzel or a (3n+1,3n+2)-benzel using left stones and two kinds of bones; see Defant et al., below. - James Propp, Jul 22 2023
a(n) is the number of labeled histories for a labeled topology with the modified lodgepole shape and n+1 cherry nodes. - Noah A Rosenberg, Jan 16 2025

Examples

			The following permutations and their reversals are all of the permutations of order 5 having the double-downgrade property:
  0 1 2 3 4
  0 3 2 1 4
  1 0 2 4 3
  1 4 2 0 3
G.f. = 1 + 2*x + 8*x^2 + 48*x^3 + 384*x^4 + 3840*x^5 + 46080*x^6 + 645120*x^7 + ...

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000142 (n!), A001147 ((2n-1)!!), A032184 (2^n*(n-1)!).
Cf. A000079, A000108, A000111, A001044, A001563, A001804, A001805, A001806.
Cf. A001807, A001813, A001907, A002454, A006882, A007060, A008277, A008544.
Cf. A010050, A019774, A028326, A028338, A035038, A039683, A047053, A047055.
Cf. A047058, A047657, A075271, A084947, A084948, A084949, A092605, A133314.
Cf. A145271, A193229, A208057.
This sequence gives the row sums in A060187, and (-1)^n*a(n) the alternating row sums in A039757.
Also row sums in A028338.
Column k=2 of A329070.

Programs

  • Haskell
    a000165 n = product [2, 4 .. 2 * n]  -- Reinhard Zumkeller, Mar 28 2015
  • Magma
    [2^n*Factorial(n): n in [0..35]]; // Vincenzo Librandi, Apr 22 2011
  • Magma
    I:=[2,8]; [1] cat [n le 2 select I[n]  else (3*n-1)*Self(n-1)-2*(n-1)^2*Self(n-2): n in [1..35] ]; // Vincenzo Librandi, Feb 19 2015
  • Maple
    A000165 := proc(n) option remember; if n <= 1 then 1 else n*A000165(n-2); fi; end;
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 0)}, labelled]: seq(combstruct[count](ZL, size=n), n=0..17); # Zerinvary Lajos, Mar 26 2008
G(x):=(1-2*x)^(-1): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..17); # Zerinvary Lajos, Apr 03 2009
A000165 := proc(n) doublefactorial(2*n) ; end proc; seq(A000165(n),n=0..10) ; # R. J. Mathar, Oct 20 2009
  • Mathematica
    Table[(2 n)!!, {n, 30}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
(2 Range[0, 30])!! (* Harvey P. Dale, Jan 23 2015 *)
RecurrenceTable[{a[n] == 2 n*a[n-1], a[0] == 1}, a, {n,0,30}] (* Ray Chandler, Jul 30 2015 *)
  • PARI
    a(n)=n!<Charles R Greathouse IV, Feb 11 2011
  • PARI
    {a(n) = prod( k=1, n, 2*k)}; /* Michael Somos, Jan 04 2013 */
  • Python
    from math import factorial
def A000165(n): return factorial(n)<Chai Wah Wu, Jan 24 2023
  • SageMath
    [2^n*factorial(n) for n in range(31)] # G. C. Greubel, Jul 21 2024

Formula

E.g.f.: 1/(1-2*x).
a(n) = A001044(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (2*i+2) = 2^n*Pochhammer(1,n). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
D-finite with recurrence a(n) = 2*n * a(n-1), n>0, a(0)=1. - Paul Barry, Aug 26 2004
This is the binomial mean transform of A001907. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
a(n) = Integral_{x>=0} x^n*exp(-x/2)/2 dx. - Paul Barry, Jan 28 2008
G.f.: 1/(1-2x/(1-2x/(1-4x/(1-4x/(1-6x/(1-6x/(1-.... (continued fraction). - Paul Barry, Feb 07 2009
a(n) = A006882(2*n). - R. J. Mathar, Oct 20 2009
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) = upper left term in M^n, M = a production matrix (twice Pascal's triangle deleting the first "2", with the rest zeros; cf. A028326):
2, 2, 0, 0, 0, 0, ...
2, 4, 2, 0, 0, 0, ...
2, 6, 6, 2, 0, 0, ...
2, 8, 12, 8, 2, 0, ...
2, 10, 20, 20, 10, 2, ...
... (End)
From Sergei N. Gladkovskii, Apr 11 2013, May 01 2013, May 24 2013, Sep 30 2013, Oct 27 2013: (Start)
Continued fractions:
G.f.: 1 + x*(Q(0) - 1)/(x+1) where Q(k) = 1 + (2*k+2)/(1-x/(x+1/Q(k+1))).
G.f.: 1/Q(0) where Q(k) = 1 + 2*k*x - 2*x*(k+1)/Q(k+1).
G.f.: G(0)/2 where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))).
G.f.: 1/Q(0) where Q(k) = 1 - x*(4*k+2) - 4*x^2*(k+1)^2/Q(k+1).
G.f.: R(0) where R(k) = 1 - x*(2*k+2)/(x*(2*k+2)-1/(1-x*(2*k+2)/(x*(2*k+2) -1/R(k+1)))). (End)
a(n) = (2n-2)*a(n-2) + (2n-1)*a(n-1), n>1. - Ivan N. Ianakiev, Aug 06 2013
From Peter Bala, Feb 18 2015: (Start)
Recurrence equation: a(n) = (3*n - 1)*a(n-1) - 2*(n - 1)^2*a(n-2) with a(1) = 2 and a(2) = 8.
The sequence b(n) = A068102(n) also satisfies this second-order recurrence. This leads to the generalized continued fraction expansion lim_{n -> oo} b(n)/a(n) = log(2) = 1/(2 - 2/(5 - 8/(8 - 18/(11 - ... - 2*(n - 1)^2/((3*n - 1) - ... ))))). (End)
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = sqrt(e) (A019774).
Sum_{n>=0} (-1)^n/a(n) = 1/sqrt(e) (A092605). (End)
Limit_{n->oo} a(n)^4 / (n * A134372(n)) = Pi. - Daniel Suteu, Apr 09 2022
a(n) = 1/([x^n] hypergeom([1], [1], x/2)). - Peter Luschny, Sep 13 2024
a(n) = Sum_{k=0..n} k!*(n-k)!*binomial(n,k)^2. - Ridouane Oudra, Jul 13 2025

A334383 Decimal expansion of Sum_{k>=0} (-1)^k/(2^k*(k!)^2).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 25 2020

Keywords

Examples

			1/(2^0*0!^2) - 1/(2^1*1!^2) + 1/(2^2*2!^2) - 1/(2^3*3!^2) + ... = 0.5591341444189799174882684679...

Links

Crossrefs

Cf. A055546, A092605.
Bessel function values: A334380 (J(0,1)), this sequence (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

Programs

  • Mathematica
    RealDigits[BesselJ[0, Sqrt[2]], 10, 110] [[1]]
  • PARI
    besselj(0, sqrt(2)) \\ Michel Marcus, Apr 26 2020

Formula

Equals BesselJ(0,sqrt(2)).
Equals BesselI(0,sqrt(2)*i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

A249455 Decimal expansion of 2/sqrt(e), a constant appearing in the expression of the asymptotic expected volume V(d) of the convex hull of randomly selected n(d) vertices (with replacement) of a d-dimensional unit cube.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Oct 29 2014

Keywords

Examples

			1.21306131942526684720759906998236090688383627...

References

  • Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 634.

Links

Crossrefs

Cf. A019774, A092605, A249456.
Cf. A074962, A243262, A243263.

Programs

Formula

Lim_{d -> infinity} V(d) =
0 if n(d) <= (2/sqrt(e) - epsilon)^d
1 if n(d) >= (2/sqrt(e) + epsilon)^d.
Equals Product_{m>=1} A(2*m)^((-1)^(m+1)*Pi^(2*m)/(2*m)!), where A(k) is the k-th generalized Glaisher-Kinkelin (or Bendersky-Adamchik) constant (A074962, A243262, A243263, ...) (Perkins and Van Gorder, 2019). - Amiram Eldar, Feb 08 2024

A027616 Number of permutations of n elements containing a 2-cycle.

Original entry on oeis.org

0, 0, 1, 3, 9, 45, 285, 1995, 15855, 142695, 1427895, 15706845, 188471745, 2450132685, 34301992725, 514529890875, 8232476226975, 139952095858575, 2519137759913775, 47863617438361725, 957272348112505425, 20102719310362613925, 442259824841726816925, 10171975971359716789275
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Links

Crossrefs

Cf. A000266, A088436, A114320.
Column k=2 of A293211.

Programs

  • Magma
    A027616:= func< n | Factorial(n)*(1- (&+[(-1/2)^j/Factorial(j): j in [0..Floor(n/2)]]) ) >;
[A027616(n): n in [0..30]]; // G. C. Greubel, Aug 05 2022
  • Maple
    S:= series((1-exp(-x^2/2))/(1-x), x, 101):
seq(coeff(S,x,j)*j!,j=0..100); # Robert Israel, May 12 2016
  • Mathematica
    nn=30; Table[n!,{n,0,nn}]-Range[0,nn]!CoefficientList[Series[Exp[-x^2/2]/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 20 2012 *)
  • PARI
    a(n) = n! * (1 - sum(k=0,floor(n/2), (-1)^k / (2^k * k!) ) );
/* Joerg Arndt, Oct 20 2012 */
  • PARI
    N=33; x='x+O('x^N);
v=Vec( 'a0 + serlaplace( (1-exp(-x^2/2))/(1-x) ) );
v[1]-='a0;  v
/* Joerg Arndt, Oct 20 2012 */
  • SageMath
    def A027616(n): return factorial(n)*(1-sum((-1/2)^k/factorial(k) for k in (0..(n//2))))
[A027616(n) for n in (0..30)] # G. C. Greubel, Aug 05 2022

Formula

E.g.f.: (1 - exp(-x^2/2)) / (1-x).
a(n) = n! * ( 1 - Sum_{k=0..floor(n/2)} (-1)^k / (2^k * k!) ).
a(n) + A000266(n) = n!. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003
Limit_{n -> oo} a(n)/n! = 1 - e^(-1/2) = 1 - A092605. - Michel Marcus, Aug 08 2013

Extensions

Added more terms, Geoffrey Critzer, Oct 20 2012

A290506 Decimal expansion of 1 - 1/e^(1/2).

Original entry on oeis.org

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

Views

Author

Arkadiusz Wesolowski, Aug 04 2017

Keywords

Comments

The amount of time that a customer has to wait for his order at some restaurant is a random variable having an exponential distribution with a mean of x minutes. The probability that the waiting time will be x/2 minutes or less is 1 - 1/e^(1/2).
Siegel's conjecture: this constant gives the density of irregular primes among all the primes (see Ribenboim and Siegel). - Stefano Spezia, Apr 22 2025

Examples

			0.3934693402873665763962004650088195465580818645128130443171078412649434...

References

  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 225.
  • C. L. Siegel, Zu zwei Bemerkungen Kummers. Nachr. Akad. d. Wiss. Göttingen, Math. Phys. Kl., II, 1964, 51-62. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan and H. Maas), Vol. III, 436-442. Springer-Verlag, Berlin, 1966.

Crossrefs

Cf. A092605.

Programs

  • Magma
    SetDefaultRealField(RealField(105)); n:=1-Exp(-1)^(1/2); Reverse(Intseq(Floor(10^105*n)));
  • Mathematica
    RealDigits[N[1 - 1/E^(1/2), 105]][[1]]
  • PARI
    1-exp(-1)^(1/2)

Formula

Equals Integral_{x = 0..1/2} exp(-x) dx.
From Amiram Eldar, Aug 24 2020: (Start)
Equals Sum_{k>=1} (-1)^(k+1)/(2^k * k!).
Equals 1 - A092605. (End)

A306858 Decimal expansion of 1 - 1/(1*3) + 1/(1*3*5) - 1/(1*3*5*7) + ...

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jun 24 2019

Keywords

Examples

			0.7247784590070763318182279676062161663121329...

Crossrefs

Cf. A001147, A014481, A060196, A068996, A092605, A113014.

Programs

  • Mathematica
    RealDigits[Sqrt[Pi/(2 Exp[1])] Erfi[1/Sqrt[2]], 10, 110] [[1]]
RealDigits[Sqrt[2] DawsonF[1/Sqrt[2]], 10, 110] [[1]]

Formula

Equals sqrt(Pi/(2*exp(1)))*erfi(1/sqrt(2)), where erfi is the imaginary error function.
Equals (1/sqrt(e)) * Sum_{k>=0} 1/(2^k * k! * (2*k+1)) = 1/(sqrt(e)) * Sum_{k>=0} 1/A014481(k). - Amiram Eldar, Nov 12 2021
Equals 1/(1+A113014). - Jon Maiga, Nov 12 2021

A103647 Decimal expansion of area of the largest rectangle under the normal curve.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Feb 18 2005

Keywords

Comments

The normal curve is 'nc' = 1/(sqrt(2*Pi))*e^(-1/2*x^2). Area = 2*x*nc. d(Area)/dx = (sqrt(2/Pi) - sqrt(2/Pi)*x^2)*e^(-1/2*x^2). Maximum at x = 1.
Occurs in a formula estimating the error in approximating a binomial distribution with a Poisson distribution. See [Prohorov]. - Eric M. Schmidt, Feb 26 2014

Examples

			0.48394144903828669959566038587112130965734394147487005097511016856...

References

  • R. E. Larson, R. P. Hostetler & B. H. Edwards, Calculus of a Single Variable, 5th Edition, D. C. Heath and Co., Lexington, MA Section 5.4 Exponential Functions: Differentiation and Integration, Exercise 61, page 351.
  • Yu. V. Prohorov, Asymptotic behavior of the binomial distribution. 1961. Select. Transl. Math. Statist. and Probability, Vol. 1 pp. 87-95. Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I.

Links

Crossrefs

Cf. A001113, A092605.

Programs

  • Mathematica
    RealDigits[ Sqrt[2/(E*Pi)], 10, 111][[1]]

Formula

Equals sqrt(2/Pi)*e^(-1/2).

A181180 Decimal expansion of exp(exp(-1/2)).

Original entry on oeis.org

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

Views

Author

Geoffrey Caveney, Oct 09 2010

Keywords

Comments

The real number y such that y = exp(xy^-x) is a maximum, at x=exp(1/2).
y=1.8340573791984...

Crossrefs

Cf. A092605 (decimal expansion of exp(-1/2)), A019774 (decimal expansion of exp(1/2)). - Klaus Brockhaus, Oct 09 2010

Programs

Extensions

More terms from Klaus Brockhaus, Oct 09 2010

A336284 Decimal expansion of Sum_{n>=2} n^(log(n))/log(n)^n.

Original entry on oeis.org

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

Views

Author

Bernard Schott, Jul 17 2020

Keywords

Comments

This series is convergent because there exists n_1 such that for n >= n_1, n^(log(n))/log(n)^n <= (1/sqrt(e))^n.

Examples

			10.5417051152289715912697153360630929474717489965883...

Crossrefs

Cf. A073009 (1/n^n), A099870 (1/n^log(n)), A099871 (1/log(n)^n), A308915 (1/(log(n)^log(n))).
Cf. A092605 (1/sqrt(e)).

Programs

  • Maple
    evalf(sum(n^(log(n))/log(n)^n, n=2..infinity),100);
  • PARI
    suminf(n=2, n^(log(n))/log(n)^n) \\ Michel Marcus, Jul 17 2020

Formula

Equals Sum_{n>=2} n^(log(n))/log(n)^n.

A093814 Decimal expansion of sqrt(2*Pi/e).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, May 20 2004

Keywords

Comments

Arises in an asymptotic formula for f(x) = Sum_{k>0} (x/k)^k as x->oo: f(x) is asymptotic to sqrt(2*Pi/e)*sqrt(x)*e^(x/e).

Links

Crossrefs

Equals A019727*A092605. - Michel Marcus, Oct 02 2018

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(2*Pi(R)/Exp(1)); // G. C. Greubel, Oct 01 2018
  • Mathematica
    RealDigits[Sqrt[2*Pi/E],10,120][[1]] (* Harvey P. Dale, Mar 05 2015 *)
  • PARI
    default(realprecision, 100); sqrt(2*Pi/exp(1)) \\ G. C. Greubel, Oct 01 2018

Formula

sqrt(2*Pi/e) = 1.52034690106628080561194...
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