A019774 -id:A019774 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

			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 + ...

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).

T. D. Noe, Table of n, a(n) for n = 0..100
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
CombOS - Combinatorial Object Server, Generate colored permutations
R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015.
Colin Defant, Rupert Li, James Propp, and Benjamin Young, Tilings of Benzels via the Abacus Bijection, arXiv preprint, arXiv:2209.05717 [math.CO], 2022.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
G. Gordon, The answer is 2^n*n! What is the question?, Amer. Math. Monthly, 106 (1999), 636-645.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Hamed Hatami and Pooya Hatami, Perfect dominating sets in the Cartesian products of prime cycles, arXiv:math/0701018 [math.CO], 2006-2009.
Jason D. Hildebrand, Differentiating Arctan(x)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 136
E. Lappo and N. A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
Eugene McDonnell, Magic Squares and Permutations, APL Quote Quad 7.3 (Fall 1976).
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.
G. A. Miller, Groups formed by special matrices, Bull. Amer. Math. Soc. 24 (1918), 203-206.
R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Eric Weisstein's World of Mathematics, Double Factorial
Eric Weisstein's World of Mathematics, Graph Automorphism
Eric Weisstein's World of Mathematics, Ladder Rung Graph
Index to divisibility sequences
Index entries for sequences related to factorial numbers

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.

a000165 n = product [2, 4 .. 2 * n]  -- Reinhard Zumkeller, Mar 28 2015

[2^n*Factorial(n): n in [0..35]]; // Vincenzo Librandi, Apr 22 2011

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

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

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 *)

a(n)=n!<Charles R Greathouse IV, Feb 11 2011

{a(n) = prod( k=1, n, 2*k)}; /* Michael Somos, Jan 04 2013 */

from math import factorial
def A000165(n): return factorial(n)<Chai Wah Wu, Jan 24 2023

[2^n*factorial(n) for n in range(31)] # G. C. Greubel, Jul 21 2024

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

A010844 a(n) = 2na(n-1) + 1 with a(0) = 1.

Original entry on oeis.org

1, 3, 13, 79, 633, 6331, 75973, 1063623, 17017969, 306323443, 6126468861, 134782314943, 3234775558633, 84104164524459, 2354916606684853, 70647498200545591, 2260719942417458913, 76864478042193603043, 2767121209518969709549, 105150605961720848962863

Offset: 0

Simon Plouffe

Related to Incomplete Gamma Function at 1/2. - Michael Somos, Mar 26 1999
For positive n, a(n) is equal to 2^n times the permanent of the n X n matrix with 3/2's along the main diagonal, and 1's everywhere else. - John M. Campbell, Jul 09 2011
Number of ways to sort a spreadsheet with n columns. (A subset of columns is chosen to sort on. These columns are ordered from major to minor, and each designated as to whether to sort by ascending or descending order. For example a spreadsheet with columns A,B,C,D could be sorted by column D ascending, then by column B descending, or any of 632 other ways.) - Marc LeBrun, Dec 07 2013
a(n) is a specific instance of sequences having the form b(0) = x, b(n) = a*n*b(n-1) + k for n >= 1. (Here x = 1, a = 2, and k = 1). Sequences of this form have a closed form of b(n) = n!*a^n*x + k*Sum_{j=1..n} n!*a^(n-j)/j!. - Gary Detlefs, Mar 26 2018

			a(3) = 2*3*a(2) + 1 = 6*13 + 1 = 79.
G.f. = 1 + 3*x + 13*x^2 + 79*x^3 + 633*x^4 + 6331*x^5 + 75973*x^6 + 1063623*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, Discrete Math., 309 (2009), 6245-6254. [From _N. J. A. Sloane_, Nov 27 2009]
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
Guo-Niu Han, Enumeration of Standard Puzzles [Wayback Machine link added by _Felix Fröhlich_, Mar 26 2018]
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
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. A000165, A000354, A000522, A007566, A010845, A019774, A097817, A097819.

G:=(x,a,k,n)-> n!*a^n*x + k*sum(n!*a^(n-j)/j!,j=1..n); seq(G(1,2,1,n), n = 0..20) # Gary Detlefs, Mar 26 2018
a := n -> 2^n*add((n!/k!)*(1/2)^k, k=0..n):
seq(a(n), n=0..19); # Peter Luschny, Jan 06 2020
seq(simplify(2^n*KummerU(-n, -n, 1/2)), n = 0..19); # Peter Luschny, May 10 2022

Table[ Gamma[ n, 1/2 ]*Exp[ 1/2 ]*2^(n-1), {n, 1, 24} ]
   and/or... s=1;lst={};Do[s+=s++n;AppendTo[lst, s], {n, 1, 5!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 23 2008 *)
a[ n_] := If[ n<0, 0, Floor[ n! E^(1/2) 2^n ]] (* Michael Somos, Sep 04 2013 *)
nxt[{n_,a_}]:={n+1,2*a(n+1)+1}; NestList[nxt,{0,1},20][[All,2]] (* Harvey P. Dale, Jan 06 2022 *)
a[n_] := n! 2^n Hypergeometric1F1[-n, -n, 1/2];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Jul 28 2024 *)

{a(n) = if( n<0, 0, n! * sum(k=0, n, 2^(n-k) / k!))} /* Michael Somos, Sep 04 2013 */

a(n) = floor(n! * e^(1/2) * 2^n) = n! * Sum_{k=0..n} 2^(n-k) / k! (i.e. binomial transform of (2n)!! = n!*2^n) = n! * (e^(1/2) * 2^n - Sum_{k >= n+1} 2^(n-k) / k!). - Michael Somos, Mar 26 1999
a(n) = A056541(n) + A000165(n). - Henry Bottomley, Jun 20 2000
E.g.f.: exp(x)/(1 - 2*x). - Vladeta Jovovic, Aug 11 2002
Sum_{n >= 1} 1/a(n) = 0.4246665348160769533082551230... - Cino Hilliard, Aug 19 2003
a(n) = Sum_{k=0..n} P(n, k)*2^k, where P(n,k) = n!/(n-k)!. - Ross La Haye, Aug 29 2005
G.f.: 1/(1 - x - 2*x/(1 - 2*x/(1 - x - 4*x/(1 - 4*x/(1 - x - 6*x/(1 - 6*x/(1 - x - 8*x/(1 - 8*x/(1 - x - 10*x/(1 - ... (continued fraction).
G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+3) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = Sum_{k=0..n} C(n,k)*k!*2^k. - Marc LeBrun, Dec 07 2013
0 = a(n)*(2*a(n+1) - 5*a(n+2) + a(n+3)) + a(n+1)*(a(n+1) + a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n > -2. - Michael Somos, Jan 02 2014
a(n) + (-2*n-1)*a(n-1) + 2*(n-1)*a(n-2) = 0. - R. J. Mathar, Jan 31 2014
a(n) = hypergeometric_U(1, n+2, 1/2)/2. - Peter Luschny, Nov 26 2014
From Peter Bala, Jan 30 2015: (Start)
a(n) = Integral_{x >= 0} (2*x + 1)^n*exp(-x) dx. (Cf. A000354.)
The e.g.f. y = exp(x)/(1 - 2*x) satisfies the differential equation (1 - 2*x)*y' = (3 - 2*x)*y. R. J. Mathar's recurrence above follows easily from this.
The sequence b(n) := 2^n*n! also satisfies R. J. Mathar's recurrence with b(0) = 1 and b(1) = 2. This leads to the continued fraction representation a(n) = 2^n*n!*( 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/(2*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/2) = 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/((2*n + 1) - ... )))). (End)
a(n) = 2^n*KummerU(-n, -n, 1/2). - Peter Luschny, May 10 2022
a(n) = n!*2^n*hypergeom([-n], [-n], 1/2). - Peter Luschny, Jul 28 2024

Better description and formulas from Michael Somos

A019645 Decimal expansion of sqrt(Pi*e).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.9222823653222778645416230761076823153979...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A019609.

C := ComplexField(); [Sqrt(Pi(C)*Exp(1))]; // G. C. Greubel, Nov 17 2017

Mathematica

RealDigits[Sqrt[Pi E],10,120][[1]] (* Harvey P. Dale, Jun 14 2014 *)

PARI

sqrt(Pi*exp(1)) \\ G. C. Greubel, Nov 17 2017

Formula

Equals A002161*A019774. - R. J. Mathar, Apr 11 2024

A025168 Expansion of e.g.f.: exp(x/(1-2*x)).

Original entry on oeis.org

1, 1, 5, 37, 361, 4361, 62701, 1044205, 19748177, 417787921, 9770678101, 250194150581, 6959638411705, 208919770666777, 6729933476435261, 231512615111396221, 8469125401589550241, 328241040596380393505, 13434223364220816489637, 578931271898150002093381

Offset: 0

Wouter Meeussen

From Peter Bala, Nov 21 2017: (Start)
The sequence terms have the form 4*m + 1 (follows from the recurrence).
For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k (proof by induction on n making use of the recurrence - the starting case a(k) == a(0) (mod k) for all k follows from the sum formula for a(k)). Hence for each k, the sequence b(n) == a(n) (mod k) is periodic with the exact period dividing k. (End)
Compound Poisson distribution with parameter 1 and distribution Geometric(1/2) has a probability mass function p_n = a(n)*e^(-1/2)/(4^n*n!). More specifically, let S = Sum_{i=0..N} X_i where X_i's are i.i.d. random variables with Geometric(1/2) distribution (i.e., Pr{X_i = k} = 1/2^(k+1) for k=0,1,2...) and N is a random variable with Poisson(1) distribution independent of all X_i's. Then Pr{S=n} = a(n)*e^(-1/2)/(4^n*n!) = a(n)*e^(-1/2)/A047053(n) for nonnegative integers n. - Xiaohan Zhang, Nov 16 2022

Robert Israel, Table of n, a(n) for n = 0..400
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003; J. Phys. A 37 (2004), 3475-3487.
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
Thomas Wieder, Expanded definitions of A103446 and A025168

Cf. A000012, A000262, A103446.

with(combstruct); SetSeqSeqL := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Sequence(Z,card >=1)},labeled];
f:= gfun:-rectoproc({a(n) = (4*n-3)*a(n-1) - 4*(n-2)*(n-1)*a(n-2),a(0)=1,a(1)=1},a(n),remember):
map(f, [$0..30]); # Robert Israel, Nov 21 2017

Table[ n! 2^n LaguerreL[ n, 1, -1/2 ], {n, 0, 12} ]
With[{nn=20},CoefficientList[Series[Exp[x/(1-2x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2012 *)

A025168 = lambda n: hypergeometric([-n,-n+1], [], 2)
[Integer(A025168(n).n(100)) for n in range(20)] # Peter Luschny, Sep 22 2014

Second LAH transform of A000012. LAH transform of A000262. a(n) = Sum_{k=0..n} 2^(n-k)*n!/k!*binomial(n-1, k-1). - Vladeta Jovovic, Oct 17 2003
Define f_1(x), f_2(x), ... such that f_1(x) = e^x, f_{n+1}(x) = (d/dx)(x^2*f_n(x)), for n=2,3,.... Then a(n) = e^(-1/2)*4*(n-1)*f_n(1/2). - Milan Janjic, May 30 2008
From Vaclav Kotesovec, Jun 22 2013: (Start)
D-finite with recurrence: a(n) = (4*n-3)*a(n-1) - 4*(n-2)*(n-1)*a(n-2).
a(n) ~ 2^(n-3/4)*n^(n-1/4)*exp(sqrt(2*n)-n-1/4) * (1-1/(3*sqrt(2*n))).
(End)
E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - x/(x + (k+1)*(1-2*x)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 09 2013
a(n) = hypergeometric([-n,-n+1],[],2). - Peter Luschny, Sep 22 2014
Sum_{n>=0} a(n)/(4^n*n!) = sqrt(e) = A019774. -Xiaohan Zhang, Nov 16 2022

Corrected and extended by Vladeta Jovovic, Sep 08 2002

A092605 Decimal expansion of e^(-1/2) or 1/sqrt(e).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

For x = e^(-1/2), the largest prime factor of a random integer n is equally likely to be above or below n^x. - Charles R Greathouse IV, May 25 2009

Siegel's conjecture: this constant gives the density of regular primes among all the primes (see Ribenboim and Siegel). - Stefano Spezia, Apr 22 2025

			0.6065306597126334...

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.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 502.
Index entries for transcendental numbers.

Cf. A000165, A001113, A019774, A068985, A290506.

RealDigits[E^-(1/2),10,120][[1]] (* Harvey P. Dale, Jul 23 2012 *)

exp(-.5) \\ Charles R Greathouse IV, Oct 02 2022

Equals Sum_{k>=0} (-1)^k/(2^k * k!) = Sum_{k>=0} (-1)^k/A000165(k). - Amiram Eldar, Aug 15 2020
From Peter Bala, Jan 16 2022; (Start)
Equals 16*Sum_{n >= 0} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9)*(2^n)*n!).
Equals 8*Sum_{n >= 0} (-1)^n/(p(n)*p(n+1)*(2^n)*n!), where p(n) = 4*n^2 + 8*n + 1.
Equals 48*Sum_{n >= 0} (-1)^n/(q(n)*q(n+1)*(2^n)*n!), where q(n) = 8*n^3 + 36*n^2 + 34*n + 1. (End)
Equals i^(i/Pi), where i denotes the imaginary unit. - Stefano Spezia, Mar 01 2025
Equals 1 - A290506. - Amiram Eldar, Apr 22 2025

A092553 Decimal expansion of 1/e^2.

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

Consider a substrate (such as polyvinyl alcohol or in forming the polymer of methyl vinyl ketone) in a "1,3 configuration" in which substituents branching off the substrate can irreversibly join with neighboring substituents unless the neighbor is already joined to its other neighbor. Then this constant is the fraction of joined substituents on an infinite substrate.
This also applies to reversible reactions when the rate of forward reaction is much faster than that of backward reaction; see Flory p. 1518 footnote 5. This had "satisfactory accord" with his experimental data using methyl vinyl ketone polymer for which the experimentally-obtained percentage was 0.15.
(A 1,k configuration is a substituent branching off a carbon atom, k-2 intermediate carbon atoms, and substituent branching off a carbon atom.)  - Charles R Greathouse IV, Nov 30 2012
Also the probability, as n increases without bound, that a permutation of length n is simple: no intervals of length 1 < k < n (an interval of a permutation s is a set of contiguous numbers which in s have consecutive indices). - Charles R Greathouse IV, May 14 2014

			0.1353352832366...

M. H. Albert, M. D. Atkinson and M. Klazar, The enumeration of simple permutations, J. Integer Seq. 6 (2003) 03.4.4. arXiv:math/0304213.
R. Brignall, A survey of simple permutations, Permutation Patterns, ed. S. Linton, N. Ruškuc and V. Vatter, Cambridge Univ. Press, 2010, pp. 41—65; arXiv:0801.0963.
Paul J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers, Journal of the American Chemical Society 61:6 (1939), pp. 1518-1521.
Index entries for transcendental numbers

Cf. A019774, A001113, A068985, A219863.

RealDigits[N[1/E^2,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

exp(-2) \\ Charles R Greathouse IV, Nov 30 2012

From Peter Bala, Oct 27 2019: (Start)
1/e^2 = Sum_{k >= 0} (-2)^k/k!.
This is the case n = 0 of the following series acceleration formulas:
1/e^2 = n!*2^n*Sum_{k >= 0} (-2)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*2^(n-k)*binomial(-x,k) are the (unsigned) row polynomials of A137346. Cf. A094816. (End)

A064618 Stirling transform of (n!)^2.

Original entry on oeis.org

1, 1, 5, 49, 821, 21121, 775205, 38516689, 2490976661, 203419086241, 20474978755205, 2490729330118129, 360263844701062901, 61114158974786823361, 12017074366801186956005, 2711409826920884006692369, 695820350706240448128979541, 201526362605605903609254528481

Offset: 0

Karol A. Penson, Sep 26 2001

From Thomas Wieder, Oct 21 2004: (Start)
"Also the number of hierarchies with labeled elements and labeled levels where the levels are permuted. Let l_x denote level x, e.g. l_2 is level 2. Let 1 denote an element and 2 a second element and so on. Then l_1:123 means elements 1,2 and 3 are on level 1.
"Let | indicate separation between levels. Then l_1:1|l_2:346|l_3:5 denotes a hierarchy of n=6 unlabeled elements with element 1 on level 1, elements 3,4 and 6 on level 2 and element 5 on level 3.
"E.g. for n=3 one has a(3) = 49 possible hierarchies:
"l_1:123,
"l_1:12|l_2:3, l_1:13|l_2:2, l_1:23|l_2:1,
"l_2:12|l_1:3, l_2:13|l_1:2, l_2:23|l_1:1,
"l_1:1|l_2:23, l_1:2|l_2:13, l_1:3|l_2:12,
"l_2:1|l_1:23, l_2:2|l_1:13, l_2:3|l_1:12,
"l_1:1|l_2:2|l_3:3 and further five permutations of the elements with levels fixed,
"l_3:1|l_1:2|l_2:3 and further five permutations of the elements with levels fixed,. etc., up to
"l_3:1|l_2:2|l_1:3 and further five permutations of the elements with levels fixed. this gives 1 + 6 +6 + 6*6 = 49 = a(3) possible hierarchies.
"See A001339 for the number of hierarchies with unlabeled elements and labeled levels."
(End)
Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 9, 1, 1, 5, 9, ...), with an apparent period 1, 1, 5, 9 of length 4 = phi(10) beginning at a(0). - Peter Bala, Jan 15 2018

Alois P. Heinz, Table of n, a(n) for n = 0..253

Cf. A001044.
Cf. A001339.
Cf. A242282, A242283, A019774, A320502.

a:= n-> add(Stirling2(n, k)*(k!^2), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 21 2012

Table[Sum[(k!)^2*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 10 2014 *)

/* By Vladeta Jovovic's formula: */
{a(n) = my(X=x+x*O(x^n)); n!*polcoeff( sum(m=0,n, m!*(exp(X)-1)^m), n)} /* Paul D. Hanna, Feb 15 2012 */

a(n) = Sum_{k=0..n} Stirling2(n, k)*(k!)^2.
E.g.f: hypergeom([1, 1], [], exp(x)-1). - Vladeta Jovovic, Sep 14 2003
O.g.f.: Sum_{n>=0} n!^2 * Product_{k=1..n} x/(1 - k*x). - Paul D. Hanna, Nov 25 2012
a(n) ~ exp(1/2) * (n!)^2. - Vaclav Kotesovec, May 10 2014

A233584 Coefficients of the generalized continued fraction expansion sqrt(e) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

1, 1, 1, 1, 5, 9, 17, 109, 260, 2909, 3072, 3310, 3678, 6715, 35175, 37269, 439792, 1400459, 1472451, 4643918, 5683171, 44850176, 62252861, 145631385, 154435765, 371056666, 1685980637, 11196453405, 14795372939

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on Blazys' expansions, see A233582.

Compared with simple continued fraction expansion for sqrt(e), this sequence starts soon growing very rapidly.

Stanislav Sykora, Table of n, a(n) for n = 1..1000
S. Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
S. Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Cf. A019774 (sqrt(e)), A058281 (simple continued fraction).

Cf. Blazys' expansions: A233582 (Pi), A233583, A233585, A233586, A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.

BlazysExpansion[n_, mx_] := Block[{k = 1, x = n, lmt = mx + 1, s, lst = {}}, While[k < lmt, s = Floor[x]; x = 1/(x/s - 1); AppendTo[lst, s]; k++]; lst]; BlazysExpansion[Sqrt@E, 35] (* Robert G. Wilson v, May 22 2014 *)

bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
bx(exp(1/2), 100) \\ Execution; use high real precision

sqrt(e) = 1+1/(1+1/(1+1/(1+1/(5+5/(9+9/(17+17/(109+...))))))).

A058281 Continued fraction for square root of e.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, 21, 1, 1, 25, 1, 1, 29, 1, 1, 33, 1, 1, 37, 1, 1, 41, 1, 1, 45, 1, 1, 49, 1, 1, 53, 1, 1, 57, 1, 1, 61, 1, 1, 65, 1, 1, 69, 1, 1, 73, 1, 1, 77, 1, 1, 81, 1, 1, 85, 1, 1, 89, 1, 1, 93, 1, 1, 97, 1, 1, 101, 1, 1, 105, 1, 1, 109, 1, 1, 113, 1, 1

Offset: 0

Robert G. Wilson v, Dec 07 2000

			sqrt(e) = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(5 + ...)))). - _Harry J. Smith_, May 01 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Keith Matthews, Finding the continued fraction of e^(l/m).
Thomas J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
Gang Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A004766, A016813.

Cf. A019774 (decimal expansion of sqrt(e)).

ContinuedFraction[ Sqrt[E], 100]
LinearRecurrence[{0,0,2,0,0,-1},{1,1,1,1,5,1},100] (* Harvey P. Dale, Aug 05 2025 *)

PARI
```
contfrac(sqrt(exp(1)))
```

{ allocatemem(932245000); default(realprecision, 60000); x=contfrac(sqrt(exp(1))); for (n=1, 20001, write("b058281.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 01 2009

a(3k+1) = 4k+1, a(i) = 1 otherwise.
G.f.: -(x^2-x+1)*(x^3-2*x^2-2*x-1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Jun 24 2013
E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 4*x) + (4 + 8*x)*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, May 05 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*log(sqrt(2)+1)) / (4*sqrt(2)). - Amiram Eldar, May 03 2025

More terms from Jason Earls, Jul 10 2001

A092041 Decimal expansion of cube root of e.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 27 2004

e^(1/3) maximizes the value of x^(c/(x^3)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

			1.39561242508608952862812531960258683759790651519940...

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

Cf. A001113, A019774, A091933, A092615 (reciprocal).

evalf(root[3](exp(1))) ; # R. J. Mathar, Oct 03 2011

RealDigits[E^(1/3), 10, 100][[1]] (* Alonso del Arte, Sep 13 2011 *)

exp(1/3) \\ Charles R Greathouse IV, Sep 14 2011

Equals (729/1552)*(1 + Sum_{n>=1} (1 + n^5/3 + n/3)/(3^n*n!)). - Alexander R. Povolotsky, Sep 13 2011
Equals (1/2)*(1 + (4 + (7 + (10 + ...)/9)/6)/3) = 1 + (1 + (1 + (1 + ...)/9)/6)/3. - Rok Cestnik, Jan 19 2017
Equals lim_{x->0} (tan(x)/x)^(1/x^2). - Amiram Eldar, Jul 04 2022

cp's OEIS Frontend

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

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A010844 a(n) = 2*n*a(n-1) + 1 with a(0) = 1.

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A019645 Decimal expansion of sqrt(Pi*e).

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A025168 Expansion of e.g.f.: exp(x/(1-2*x)).

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A092605 Decimal expansion of e^(-1/2) or 1/sqrt(e).

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A092553 Decimal expansion of 1/e^2.

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A010844 a(n) = 2na(n-1) + 1 with a(0) = 1.