A002454 -id:A002454 - 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

A001818 Squares of double factorials: (135*...(2n-1))^2 = ((2n-1)!!)^2.

Original entry on oeis.org

1, 1, 9, 225, 11025, 893025, 108056025, 18261468225, 4108830350625, 1187451971330625, 428670161650355625, 189043541287806830625, 100004033341249813400625, 62502520838281133375390625, 45564337691106946230659765625, 38319607998220941779984862890625

Offset: 0

N. J. A. Sloane

Number of permutations in S_{2n} in which all cycles have even length (cf. A087137).
Also number of permutations in S_{2n} in which all cycles have odd length. - Vladeta Jovovic, Aug 10 2007
a(n) is the sum over all multinomials M2(2*n,k), k from {1..p(2*n)} restricted to partitions with only even parts. p(2*n)= A000041(2*n) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n,k). - Wolfdieter Lang, Aug 07 2007
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture 1: For any primitive 2n-th root zeta of unity, the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} coincides with a(n) = ((2n-1)!!)^2, where m(j,k) is (1+zeta^(j-k))/(1-zeta^(j-k)) if j is not equal to k, and 1 otherwise.
The determinant of [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n-1)!!)^2/(2n-1) by Han Wang and Zhi-Wei Sun in 2022.
Conjecture 2: Let p be an odd prime. Then the permanent of (p-1) X (p-1) matrix [f(j,k)]_{j,k=1..p-1} is congruent to a((p-1)/2) = ((p-2)!!)^2 modulo p^2, where f(j,k) is (j+k)/(j-k) if j is not equal to k, and f(j,k) = 1 otherwise. (End)

			Multinomial representation for a(2): partitions of 2*2=4 with even parts only: (4) with position k=1, (2^2) with k=3; M2(4,1)= 6 and M2(4,3)= 3, adding up to a(2)=9.
G.f. = 1 + x + 9*x^2 + 225*x^3 + 11025*x^4 + 893025*x^5 + 108056025*x^6 + ...

John Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.34(c).

T. D. Noe, Table of n, a(n) for n = 0..50
Ron M. Adin, Pál Hegedűs, and Yuval Roichman, Descent set distribution for permutations with cycles of only odd or only even lengths, arXiv:2502.03507 [math.CO], 2025. See p. 2.
David Callan and Emeric Deutsch, The Run Transform, arXiv preprint arXiv:1112.3639 [math.CO], 2011.
William Y. C. Chen and Elena L. Wang, r-Enriched Permutations and an Inequality of Bóna-McLennan-White, arXiv:2502.04136 [math.CO], 2025. See p. 4.
Harry Crane and Peter McCullagh, Reversible Markov structures on divisible set partitions, Journal of Applied Probability, Vol. 52, No. 3 (2015), pp. 622-635.
Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024. See p. 11.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 6.
IBM, "Ponder This" puzzle for June 2009. [From _Vladeta Jovovic_, Jul 26 2009]
John Riordan and N. J. A. Sloane, Correspondence, 1974.
Terence Tao, A differentiation identity.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Struve function.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
Index to divisibility sequences.

Cf. A001147, A002454, A111595, A197037.
Bisection of A012248.
Right-hand column 1 in triangle A008956.

DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((2*n-1))^2: n in [0..20] ]; // Vincenzo Librandi, Jul 21 2017

a := proc(m) local k; 4^m*mul((-1)^k*(k-m-1/2),k=1..2*m) end; # Peter Luschny, Jun 01 2009

FoldList[Times,1,Range[1,25,2]]^2 (* or *) Join[{1},(Range[1,29,2]!!)^2] (* Harvey P. Dale, Jun 06 2011, Apr 10 2012 *)
Table[((2 n - 1)!!)^2, {n, 0, 30}] (* Vincenzo Librandi, Jul 21 2017 *)

PARI
```
a(n)=((2*n)!/(n!*2^n))^2
```

{a(n) = if( n<0, 1 / a(-n), sqr((2*n)! / (n! * 2^n)))}; /* Michael Somos, Jan 06 2017 */

a(n) = A001147(n)^2.
a(n) = A111595(2*n, 0).
a(n) = (2*n-1)!*Sum_{k=0..n-1} binomial(2*k,k)/4^k, n >= 1. - Wolfdieter Lang, Aug 23 2005
arcsinh(x) = Sum_{n>=1} (-1)^(n-1)*a(n)*x^(2*n-1)/(2*n-1)!. - James R. Buddenhagen, Mar 24 2009
From Karol A. Penson, Oct 21 2009: (Start)
G.f.: Sum_{n>=0} a(n)*x^n/(n!)^2 = 2*EllipticK(2*sqrt(x))/Pi.
Asymptotically: a(n) = (2/((exp(-1/2))^2*(exp(1/2))^2)-1/(6*(exp(-1/2))^2*(exp(1/2))^2*n)+1/(144*(exp(-1/2))^2*(exp(1/2))^2*n^2)+O(1/n^3))*(2^n)^2/(((1/n)^n)^2*(exp(n))^2), n->infinity.
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:
a(n) = Integral_{x>=0} x^n*BesselK(0,sqrt(x))/(Pi*sqrt(x)).
This solution is unique.
(End)
D-finite with recurrence: a(0) = 1, a(n) = (2*n-1)^2*a(n-1), n > 0.
a(n) ~ 2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: 1/sqrt(1-x^2) = Sum_{n >= 0} a(n)*x^(2*n)/(2*n)!. Also arcsin(x) = Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)!. - Michael Somos, Jul 03 2002
(-1)^n*a(n) is the coefficient of x^0 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22 2002
-arccos(x) + Pi/2 = x + x^3/3! + 9*x^5/5! + 225*x^7/7! + 11205*x^9/9! + ... - Tom Copeland, Oct 23 2008
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) =  1 - (4*k^2+4*k+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = det(V(i+1,j), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices. - Mircea Merca, Apr 04 2013
a(n) = (1+x^2)^(n+1/2) * (d/dx)^(2*n) (1+x^2)^(n-1/2).  See Tao link. - Robert Israel, Jun 04 2015
a(n) = 4^n * gamma(n + 1/2)^2 / Pi. - Daniel Suteu, Jan 06 2017
0 = a(n)*(+384*a(n+2) - 60*a(n+3) + a(n+4)) + a(n+1)*(-36*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) and a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 06 2017
From Robert FERREOL, Jul 30 2020: (Start)
a(n) = ((2*n)!/4^n)*binomial(2*n,n).
a(n) = (2*n-1)!*Sum_{k=0..n-1} a(k)/(2*k)!, n >= 1.
a(n) = A184877(2*n-1) for n>=1. (End)
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_0(1)*Pi/2, where L is the modified Struve function (see A197037).
Sum_{n>=0} (-1)^n/a(n) = 1 - H_0(1)*Pi/2, where H is the Struve function. (End)

Incorrect formula deleted by N. J. A. Sloane, Jul 03 2009

A007106 Number of labeled odd degree trees with 2n nodes.

Original entry on oeis.org

1, 4, 96, 5888, 686080, 130179072, 36590059520, 14290429935616, 7405376630685696, 4917457306800619520, 4071967909087792857088, 4113850542422629363482624, 4980673081258443273955966976, 7119048451600750435732824260608, 11861520124846917915630931846103040

Offset: 1

N. J. A. Sloane

nonn

			From _Peter Bala_, Apr 24 2012: (Start)
Let G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + ... be the e.g.f. for A143601. Then sinh(x*G(x)) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
Conjectural e.g.f. as an x-adic limit:
sinh(x) = x + ...; sinh(x*cosh(x)) = x + 4*x^3/3! + ...;
sinh(x*cosh(x*cosh(x))) = x + 4*x^3/3! + 96*x^5/5! + ...;
sinh(x*cosh(x*cosh(x*cosh(x)))) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
(End)

R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..211 (terms 1..39 from R. W. Robinson)
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees
Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees II
Marko Riedel, Count by Prüfer codes and Stirling numbers
Index entries for sequences related to trees

Cf. A036778, A058014, A060279, A143601, A156289.

A007106(n) = A(2n) where n>=2, A(n) = (add(binomial(n,q)*(n-2*q)^(n-2)/(n-2)!, q=0..n) - add(binomial(n-1,q)*(n-2*q)^(n-3)/(n-3)!, q=0..n-1) + add(binomial(n-1,q)*(n-2-2*q)^(n-3)/(n-3)!, q=0..n-1))*n!/2^(n+1)/(n-1)

{1}~Join~Array[(1/2)*Sum[Binomial[2 #, k]*(# - k)^(2 # - 2), {k, 0, # - 1}] &, 12, 2] (* Michael De Vlieger, Oct 13 2021 *)

a(n) = if(n<=1, n==1, sum(k=0, n-1, binomial(2*n,k) * (n-k)^(2*n-2))/2) \\ Andrew Howroyd, Nov 22 2021

a(n) = A060279(n)/(2*n). - Vladeta Jovovic, Feb 08 2005
Bisection of A058014. The expansion 1/sqrt(1+x^2)*arcsinh(x) = x - 4*x^3/3! + 64*x^5/5! - ... (see A002454) has series reversion x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + .... The coefficients appear to be the terms of this sequence. As an x-adic limit this e.g.f. equals lim_{n -> infinity} sinh(f(n,x)), where f(0,x) = x and f(n,x) = x*cosh(f(n-1,x)) for n >= 1. See the example section below. - Peter Bala, Apr 24 2012
a(n) = Sum_{k=1..n} binomial(n,k) * k! * (n-2)! [z^{n-2}] [u^k] exp(u(exp(z)+exp(-z)-2)/2)). - Marko Riedel, Jun 16 2016
From Alexander Burstein, Oct 13 2021: (Start)
a(n) = (1/2) * Sum_{k=0..n-1} binomial(2*n,k) * (n-k)^(2*n-2) for n >= 2.
a(n) = (2*n-1)!*[x^(2*n-1)] sinh(REVERT(x/cosh(x))), see A036778. (End)
a(n) = Sum_{k=0..n-1} A156289(n-1, k)*(2*n)!/(2*n-k)!. - Peter Luschny, May 07 2022

Corrected and extended by Vladeta Jovovic, Feb 08 2005

A197036 Decimal expansion of the Modified Bessel Function I of order 0 at 1.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Oct 08 2011

			1.26606587775200833559824462521471753760767031135496...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 51, page 504.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions, Chapter 9.6.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A002454, A242282.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), this sequence (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

Maple
```
BesselI(0,1) ;evalf(%) ;
```

RealDigits[BesselJ[0, I], 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

besseli(0,1) \\ Charles R Greathouse IV, Feb 19 2014

I_0(1) = Sum_{k>=0} 1/(4^k*k!^2) = Sum_{k>=0} 1/A002454(k).
Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t)) dt.
Equals BesselJ(0,i). - Jianing Song, Sep 18 2021
From Amiram Eldar, Jul 09 2023: (Start)
Equals exp(-1) * Sum_{k>=0} binomial(2*k,k)/(2^k*k!).
Equals e * Sum_{k>=0} (-1/2)^k * binomial(2*k,k)/k!. (End)

A002474 Denominators of coefficients of odd powers of x of the expansion of Bessel function J_1(x).

Original entry on oeis.org

2, 16, 384, 18432, 1474560, 176947200, 29727129600, 6658877030400, 1917756584755200, 690392370511872000, 303772643025223680000, 160391955517318103040000, 100084580242806496296960000, 72861574416763129304186880000, 61203722510081028615516979200000

Offset: 0

N. J. A. Sloane

The corresponding numerators are A033999(n) = (-1)^n.

			a(3) = 18432 = 128*6*24, since J_{1}(x) = x/2 - x^3/16 + x^5/384 - x^7/18432 + ...

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.4.7
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:2 and 52:6:3 at page 483, 513.

Cf. J_0: A002454, J_2: A002506, J_3: A014401, J_4: A061403, J_5: A061404, J_6: A061405, J_7: A061407, J_9: A061440 J_10: A061441.

Cf. A010790, A033999.

[2^(2*n+1)*Factorial(n)*Factorial(n+1): n in [0..30]]; // G. C. Greubel, Sep 21 2024

a:= n-> denom(coeff(series(BesselJ(1, x), x, 2*n+2), x, 2*n+1)):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 21 2024

CoefficientList[Series[BesselJ[1,x], {x,0,30}], x][[2 ;; ;; 2]]//Denominator
Table[2^(2*n+1)*n!*(n+1)!, {n,0,30}] (* G. C. Greubel, Sep 21 2024 *)

a(n) = n!^2 * (n+1) << (2*n+1) \\ Charles R Greathouse IV, Oct 23 2023

first(n)=my(x='x+O('x^(2*n+1)),t=besselj(1,x)); vector(n+1,k,2*denominator(polcoeff(t,2*k-2))) \\ Charles R Greathouse IV, Oct 23 2023

[2^(2*n+1)*factorial(n)*factorial(n+1) for n in range(31)] # G. C. Greubel, Sep 21 2024

a(n) = 2^(2n+k) * n! * (n+k)! here for k=1, i.e., Bessel's J1(x) has the denominator a(n) for the coefficient of x^(2*n+1), n >= 0.

a(n) = 2^(2n+1)*A010790(n).

Name specified, numerators given, formula augmented by Wolfdieter Lang, Aug 25 2015

A002506 Denominators of coefficients of expansion of Bessel function J_2(x).

Original entry on oeis.org

8, 96, 3072, 184320, 17694720, 2477260800, 475634073600, 119859786547200, 38355131695104000, 15188632151261184000, 7290543432605368320000, 4170190843450270679040000, 2802368246798581896314880000

Offset: 0

N. J. A. Sloane

nonn

			a(2) = 3072 = 64*2*24, J2(x) = x^2/8 - x^4/96 + x^6/3072 - x^8/184320 +- ...

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.4.7

J0: A002454, J1: A002474, J3: A014401.

Denominator[Take[CoefficientList[Series[BesselJ[2,x],{x,0,30}],x],{3,-1,2}]] (* Harvey P. Dale, Sep 21 2013 *)

a(n) = 2^(2n+k) * n! * (n+k)! here for k=2, i.e., Bessel's J2(x).

a(n) - 4*n*(n+2)*a(n-1) = 0. - R. J. Mathar, Jun 20 2013

Previous Mathematica program corrected by Harvey P. Dale, Sep 21 2013

A014401 Denominators of coefficients of expansion of Bessel function J_3(x).

Original entry on oeis.org

48, 768, 30720, 2211840, 247726080, 39636172800, 8561413324800, 2397195730944000, 843812897292288000, 364527171630268416000, 189554129247739576320000, 116765343616607579013120000

Offset: 0

N. J. A. Sloane

nonn

			a(1) = 768 = 32*24, J3(x) = x^3/48 - x^5/768 + x^7/30720 - x^9/2211840 +- ...

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7

J0: A002454, J1: A002474, J2: A002506.

k:=3: f:= gfun:-rectoproc({a(n)-(4*n^2 + 4*n*k)*a(n-1), a(0)=2^k*k!}, a(n), remember): map(f, [$0..16]); # Georg Fischer, Mar 22 2022

Denominator[Take[CoefficientList[Series[BesselJ[3,x],{x,0,30}],x],{4,-1,2}]] (* Harvey P. Dale, Dec 10 2011 *)

a(n) = 2^(2n+k) * n! * (n+k)! here for k=3, i.e., Bessel's J3(x).

D-finite with recurrence: a(n) - (4*n^2 + 4*n*k)*a(n-1) = 0, a(0) = 2^k*k!, here for k=3. - Georg Fischer, Mar 22 2022

A334380 Decimal expansion of Sum_{k>=0} (-1)^k/((2*k)!!)^2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 25 2020

This constant is transcendental.

			1/(4^0*0!^2) - 1/(4^1*1!^2) + 1/(4^2*2!^2) - 1/(4^3*3!^2) + ... = 0.765197686557966551449717526...

Cf. A000165, A002454.

Bessel function values: this sequence (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselJ[0, 1], 10, 110] [[1]]

besselj(0, 1) \\ Michel Marcus, Apr 26 2020

Equals BesselJ(0,1).

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

A001900 Successive numerators of Wallis's approximation to Pi/2 (unreduced).

Original entry on oeis.org

1, 2, 4, 16, 64, 384, 2304, 18432, 147456, 1474560, 14745600, 176947200, 2123366400, 29727129600, 416179814400, 6658877030400, 106542032486400, 1917756584755200, 34519618525593600, 690392370511872000, 13807847410237440000, 303772643025223680000

Offset: 0

N. J. A. Sloane

a(n) = number of permutations of [n+1] all of whose non-initial left-to-right minima are at even positions in the permutation. For example, a(2) = 4 counts 123, 132, 213, 312. - David Callan, Jul 22 2008
Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal. a(2) = 4: [(0,0),(1,0),(2,0)], [(0,0),(0,1),(1,0),(2,0)], [(0,0),(0,1),(0,2),(1,1),(2,0)], [(0,0),(1,0),(0,1),(0,2),(1,1),(2,0)]. - Alois P. Heinz, Mar 23 2017
a(n+1) is the number of 0-1 square matrices of order n+1 with 2n+1 nonzero entries where the cell (i,j) is 1 for all i+j=n+2 and every diagonal, parallel to the main diagonal, has exactly one 1. For example, a(2) = 4: [(0,1,1), (1,1,0), (1,0,0)], [(0,1,1), (0,1,0), (1,1,0)], [(0,0,1), (1,1,1), (1,0,0)], [(0,0,1), (0,1,1), (1,1,0)]. - Christian Barrientos, Jul 17 2021

			From _Wolfdieter Lang_, Dec 06 2017: (Start)
Partial products of the rows N (for numerators a(n)) and D (for denominators b(n) = A000246(n+1)) begin:
n:    0  1  2  3  4   5    6     7      8       9       10 ...
N:    1  2  2  4  4   6    6     8      8      10       10 ...
D:    1  1  3  3  5   5    7     7      9       9       11 ...
a(n): 1  2  4 16 64 384 2304 18432 147456 14745601 4745600 ...
b(n): 1  1  3  9 45 225 1575 11025  99225   893025 9823275 ...  (End)

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

Alois P. Heinz, Table of n, a(n) for n = 0..449
John Derbyshire, Prime Obsession, Plume books, p. 16, 2003.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Index to divisibility sequences

Cf. A000246, A284230.

For the reduced form see A001901(n)/A001902(n), n >= 0.

a[n_] := a[n] = If[n==0, 1, (n+Mod[n, 2]) a[n-1]];
a /@ Range[0, 21] (* Jean-François Alcover, Jan 31 2020 *)

a(n)=if(n<0,0,prod(k=1,n,if(k%2,k+1,k)))

2.2.4.4.6.6....2n.2n.../1.3.3.5.5.7.7....(2n-1).(2n+1) ...for n >= 1.
a(n) = 2^n * A010551(n) = 2^n * [n/2]! * [(n+1)/2]!. - Ralf Stephan, Mar 11 2004
Conjecture: a(n) - a(n-1) - n*(n-1)*a(n-2) = 0. - R. J. Mathar, Jun 07 2013 [The proof, for n >= 2, follows from the bisection recurrence given below. - Wolfdieter Lang, Dec 07 2017]
E.g.f.: E(0), where E(k)= 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)/(x + 1/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
G.f.: G(0), where G(k)= 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
Bisection: a(2*k+1) = ((2*k+1)+1)*a(2*k), a(2*k) = 2*k*a(2*k-1), k >= 0, with a(0) = 1. The proof is obvious from the numbers in the numerator (see the row N in the example). From a proposal by David James Sycamore, Nov 02 2017 based on the fractions 4/1, 8/3, 32/9, 128/45, ... converging very slowly to Pi, given on p. 16 of the Derbyshire link. - Wolfdieter Lang, Dec 06 2017
From Mathias Zechmeister, Jul 24 2022: (Start)
Let J_0(x) and J_1(x) denote the Bessel functions and i = sqrt(-1).
a(n) = denominator([x^n](J_0(x) + J_1(x))).
a(n) = denominator([x^n](J_0(i*x) - i*J_1(i*x))).
Bisection: a(2*k) = A002454(k), a(2*k+1) = A002474(k).
G.f. for 1/a(n): J_0(i*x) - i*J_1(i*x).  (End)

cp's OEIS Frontend

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

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A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.

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A007106 Number of labeled odd degree trees with 2n nodes.

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A197036 Decimal expansion of the Modified Bessel Function I of order 0 at 1.

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A002474 Denominators of coefficients of odd powers of x of the expansion of Bessel function J_1(x).

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A001818 Squares of double factorials: (135*...(2n-1))^2 = ((2n-1)!!)^2.