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 A001147 Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1). (Formerly M3002 N1217) 474

%I M3002 N1217

%S 1,1,3,15,105,945,10395,135135,2027025,34459425,654729075,13749310575,

%T 316234143225,7905853580625,213458046676875,6190283353629375,

%U 191898783962510625,6332659870762850625,221643095476699771875,8200794532637891559375

%N Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).

%C The solution to Schröder's third problem.

%C Number of fixed-point-free involutions in symmetric group S_{2n} (cf. A000085).

%C a(n+2) is the number of full Steiner topologies on n points with n-2 Steiner points.

%C a(n) is also the number of perfect matchings in the complete graph K(2n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001

%C Number of ways to choose n disjoint pairs of items from 2*n items. - Ron Zeno (rzeno(AT)hotmail.com), Feb 06 2002

%C Number of ways to choose n-1 disjoint pairs of items from 2*n-1 items (one item remains unpaired). - _Bartosz Zoltak_, Oct 16 2012

%C For n >= 1 a(n) is the number of permutations in the symmetric group S_(2n) whose cycle decomposition is a product of n disjoint transpositions. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001

%C a(n) is the number of distinct products of n+1 variables with commutative, nonassociative multiplication. - Andrew Walters (awalters3(AT)yahoo.com), Jan 17 2004. For example, a(3)=15 because the product of the four variables w, x, y and z can be constructed in exactly 15 ways, assuming commutativity but not associativity: 1. w(x(yz)) 2. w(y(xz)) 3. w(z(xy)) 4. x(w(yz)) 5. x(y(wz)) 6. x(z(wy)) 7. y(w(xz)) 8. y(x(wz)) 9. y(z(wx)) 10. z(w(xy)) 11. z(x(wy)) 12. z(y(wx)) 13. (wx)(yz) 14. (wy)(xz) 15. (wz)(xy).

%C a(n) = E(X^(2n)), where X is a standard normal random variable (i.e. X is normal with mean = 0, variance = 1). So for instance a(3) = E(X^6) = 15, etc. See Abramowitz and Stegun or Hoel, Port and Stone. - Jerome Coleman, Apr 06 2004

%C Second Eulerian transform of 1,1,1,1,1,1,... The second Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum_{k=0..n} E(n,k)s(k), where E(n,k) is a second-order Eulerian number (A008517). - _Ross La Haye_, Feb 13 2005

%C Integral representation as n-th moment of a positive function on the positive axis, in Maple notation: a(n) = int(x^n*exp(-x/2)/sqrt(2*Pi*x), x=0..infinity), n=0,1... . - _Karol A. Penson_, Oct 10 2005

%C a(n) is the number of binary total partitions of n+1 (each non-singleton block must be partitioned into exactly two blocks) or, equivalently, the number of unordered full binary trees with n+1 labeled leaves (Stanley, ex 5.2.6). - _Mitch Harris_, Aug 01 2006

%C a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is i for i<j. - _David Callan_, Sep 25 2006

%C a(n) is the number of increasing ordered rooted trees on n+1 vertices where "increasing" means the vertices are labeled 0,1,2,...,n so that each path from the root has increasing labels. Increasing unordered rooted trees are counted by the factorial numbers A000142. - _David Callan_, Oct 26 2006

%C Number of perfect multi Skolem-type sequences of order n. - _Emeric Deutsch_, Nov 24 2006

%C a(n) = total weight of all Dyck n-paths (A000108) when each path is weighted with the product of the heights of the terminal points of its upsteps. For example with n=3, the 5 Dyck 3-paths UUUDDD, UUDUDD, UUDDUD, UDUUDD, UDUDUD have weights 1*2*3=6, 1*2*2=4, 1*2*1=2, 1*1*2=2, 1*1*1=1 respectively and 6+4+2+2+1=15. Counting weights by height of last upstep yields A102625. - _David Callan_, Dec 29 2006

%C a(n) is the number of increasing ternary trees on n vertices. Increasing binary trees are counted by ordinary factorials (A000142) and increasing quaternary trees by triple factorials (A007559). - _David Callan_, Mar 30 2007

%C This sequence is essentially self-reciprocal under the list partition transform and associated operations in A133314. More precisely, A001147 and -A001147 with a leading 1 attached are reciprocal. Therefore their e.g.f.s are reciprocal. See A132382 for an extension of this result. - _Tom Copeland_, Nov 13 2007

%C From _Ross Drewe_, Mar 16 2008: (Start)

%C This is also the number of ways of arranging the elements of n distinct pairs, assuming the order of elements is significant but the pairs are not distinguishable, i.e., arrangements which are the same after permutations of the labels are equivalent.

%C If this sequence and A000680 are denoted by a(n) and b(n) respectively, then a(n) = b(n)/n! where n! = the number of ways of permuting the pair labels.

%C For example, there are 90 ways of arranging the elements of 3 pairs [1 1], [2 2], [3 3] when the pairs are distinguishable: A = { [112233], [112323], .... [332211] }.

%C By applying the 6 relabeling permutations to A, we can partition A into 90/6 = 15 subsets: B = { {[112233], [113322], [221133], [223311], [331122], [332211]}, {[112323], [113232], [221313], [223131], [331212], [332121]}, ....}

%C Each subset or equivalence class in B represents a unique pattern of pair relationships. For example, subset B1 above represents {3 disjoint pairs} and subset B2 represents {1 disjoint pair + 2 interleaved pairs}, with the order being significant (contrast A132101). (End)

%C A139541(n) = a(n) * a(2*n). - _Reinhard Zumkeller_, Apr 25 2008

%C a(n+1) = Sum_{j=0..n} A074060(n,j) * 2^j. - _Tom Copeland_, Sep 01 2008

%C From _Emeric Deutsch_, Jun 05 2009: (Start)

%C a(n) is the number of adjacent transpositions in all fixed-point-free involutions of {1,2,...,2n}. Example: a(2)=3 because in 2143=(12)(34), 3412=(13)(24), and 4321=(14)(23) we have 2 + 0 + 1 adjacent transpositions.

%C a(n) = Sum_{k>=0} k*A079267(n,k).

%C (End)

%C Hankel transform is A137592. - _Paul Barry_, Sep 18 2009

%C (1, 3, 15, 105, ...) = INVERT transform of A000698 starting (1, 2, 10, 74, ...). - _Gary W. Adamson_, Oct 21 2009

%C a(n) = (-1)^(n+1)*H(2*n,0), where H(n,x) is the probabilists' Hermite polynomial. The generating function for the probabilists' Hermite polynomials is as follows: exp(x*t-t^2/2) = Sum_{i>=0} H(i,x)*t^i/i!. - _Leonid Bedratyuk_, Oct 31 2009

%C The Hankel transform of a(n+1) is A168467. - _Paul Barry_, Dec 04 2009

%C Partial products of odd numbers. - _Juri-Stepan Gerasimov_, Oct 17 2010

%C See A094638 for connections to differential operators. - _Tom Copeland_, Sep 20 2011

%C a(n) is the number of subsets of {1,...,n^2} that contain exactly k elements from {1,...,k^2} for k=1,...,n. For example, a(3)=15 since there are 15 subsets of {1,2,...,9} that satisfy the conditions, namely, {1,2,5}, {1,2,6}, {1,2,7}, {1,2,8}, {1,2,9}, {1,3,5}, {1,3,6}, {1,3,7}, {1,3,8}, {1,3,9}, {1,4,5}, {1,4,6}, {1,4,7}, {1,4,8}, and {1,4,9}. - _Dennis P. Walsh_, Dec 02 2011

%C a(n) is the leading coefficient of the Bessel polynomial y_n(x) (cf. A001498). - _Leonid Bedratyuk_, Jun 01 2012

%C For n>0: a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j)^2 for 1 <= i,j <= n. - Enrique Pérez Herrero, Jan 14 2013

%C a(n) is also the numerator of the mean value from 0 to Pi/2 of sin(x)^(2n). - _Jean-François Alcover_, Jun 13 2013

%C a(n) is the size of the Brauer monoid on 2n points (see A227545). - _James Mitchell_, Jul 28 2013

%C For n>1: a(n) is the numerator of M(n)/M(1) where the numbers M(i) have the property that M(n+1)/M(n) ~ n-1/2 (for example, large Kendell-Mann numbers, see A000140 or A181609, as n --> infinity). - _Mikhail Gaichenkov_, Jan 14 2014

%C a(n) = the number of upper-triangular matrix representations required for the symbolic representation of a first order central moment of the multivariate normal distribution of dimension 2(n-1), i.e., E[X_1*X_2...*X_(2n-2)|mu=0, Sigma]. See vignette for symmoments R package on CRAN and Phillips reference below. - _Kem Phillips_, Aug 10 2014

%C For n>1: a(n) is the number of Feynman diagrams of order 2n (number of internal vertices) for the vacuum polarization with one charged loop only, in quantum electrodynamics. - _Robert Coquereaux_, Sep 15 2014

%C Aerated with intervening zeros (1,0,1,0,3,...) = a(n) (cf. A123023), the e.g.f. is e^(t^2/2), so this is the base for the Appell sequence A099174 with e.g.f. e^(t^2/2) e^(x*t) = exp(P(.,x),t) = unsigned A066325(x,t), the probabilist's (or normalized) Hermite polynomials. P(n,x)= (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for A066325(x,t) = exp(UP(.,x),t), i.e., UP(n,P(.,t))= x^n = P(n,UP(.,t)), where UP(n,t) are the polynomials of A066325 and, e.g., (P(.,t))^n = P(n,t). - _Tom Copeland_, Nov 15 2014

%C a(n) = the number of relaxed compacted binary trees of right height at most one of size n. A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The numer of unbounded relaxed compacted binary trees of size n is A082161(n). See the Genitrini et al. link. - _Michael Wallner_, Jun 20 2017

%C Also the number of distinct adjacency matrices in the n-ladder rung graph. - _Eric W. Weisstein_, Jul 22 2017

%C From _Christopher J. Smyth_, Jan 26 2018: (Start)

%C a(n) = the number of essentially different ways of writing a probability distribution taking n+1 values as a sum of products of binary probability distributions. See comment of Mitch Harris above. This is because each such way corresponds to a full binary tree with n+1 leaves, with the leaves labeled by the values. (This comment is due to Niko Brummer.)

%C Also the number of binary trees with root labeled by an (n+1)-set S, its n+1 leaves by the singleton subsets of S, and other nodes labeled by subsets T of S so that the two daughter nodes of the node labeled by T are labeled by the two parts of a 2-partition of T. This also follows from Mitch Harris' comment above, since the leaf labels determine the labels of the other vertices of the tree.

%C (End)

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, (26.2.28).

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228, #19.

%D Hoel, Port and Stone, Introduction to Probability Theory, Section 7.3.

%D F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, North-Holland, 1992, see p. 14.

%D C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.

%D L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 48.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.6 and also p. 178.

%D R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer-Verlag, New York, 1999, p. 73.

%H Stefano Spezia, <a href="/A001147/b001147.txt">Table of n, a(n) for n = 0..400</a> (first 102 terms from T. D. Noe)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Jonathan Burns, <a href="http://shell.cas.usf.edu/~saito/DNAweb/SimpleAssemblyTable.txt">Assembly Graph Words - Single Transverse Component (Counts)</a>.

%H Christian Aebi and Grant Cairns, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.433">Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials</a>, The American Mathematical Monthly 122.5 (2015): 433-443.

%H D. Arques and J.-F. Beraud, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00197-1">Rooted maps on orientable surfaces, Riccati's equation and continued fractions</a>, Discrete Math., 215 (2000), 1-12.

%H Fatemeh Bagherzadeh, M. Bremner, S. Madariaga, <a href="https://arxiv.org/abs/1611.01214">Jordan Trialgebras and Post-Jordan Algebras</a>, arXiv preprint arXiv:1611.01214 [math.RA], 2016.

%H Cyril Banderier, Philippe Marchal, Michael Wallner, <a href="https://arxiv.org/abs/1805.09017">Rectangular Young tableaux with local decreases and the density method for uniform random generation</a> (short version), arXiv:1805.09017 [cs.DM], 2018.

%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.html">On a Generalization of the Narayana Triangle</a>, J. Int. Seq. 14 (2011) # 11.4.5

%H Paul Barry, <a href="https://arxiv.org/abs/1805.02274">On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1805.02274 [math.CO], 2018.

%H P. Barry, A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry2/barry94r.html">The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences</a>, J. Int. Seq. 13 (2010) # 10.8.2.

%H O. Bodini, M. Dien, X. Fontaine, A. Genitrini, H. K. Hwang, <a href="https://doi.org/10.1007/978-3-662-49529-2_16">Increasing Diamonds</a>, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp 207-219 2016 DOI 10.1007/978-3-662-49529-2_16; Lecture Notes in Computer Science Series Volume 9644.

%H H. Bottomley, <a href="/A002694/a002694.gif">Illustration for A000108, A001147, A002694, A067310 and A067311</a>

%H Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, <a href="http://jtburns.myweb.usf.edu/assembly/papers/Graphs_and_DNA_Recomb_2011.pdf">Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination</a>, May 23, 2011.

%H David Callan, <a href="http://arxiv.org/abs/0906.1317">A combinatorial survey of identities for the double factorial</a>, arXiv:0906.1317 [math.CO], 2009.

%H Peter J. Cameron, <a href="http://dx.doi.org/10.1093/qmath/38.2.155">Some treelike objects</a> Quart. J. Math. Oxford Ser. 38 (1987), 155-183. MR0891613 (89a:05009). See p. 155.

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H P. Codara, O. M. D'Antona, P. Hell, <a href="http://arxiv.org/abs/1308.1700">A simple combinatorial interpretation of certain generalized Bell and Stirling numbers</a>, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.

%H Thierry Dana-Picard, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Dana-Picard/dana23.html">Sequences of Definite Integrals, Factorials and Double Factorials</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.6.

%H Filippo Disanto and Thomas Wiehe, <a href="http://arxiv.org/abs/1112.1295">Some combinatorial problems on binary rooted trees occurring in population genetics</a>, arXiv preprint arXiv:1112.1295 [math.CO], 2011.

%H John Engbers, David Galvin, Clifford Smyth, <a href="https://arxiv.org/abs/1610.05803">Restricted Stirling and Lah numbers and their inverses</a>, arXiv:1610.05803 [math.CO], 2016. See p. 5.

%H J. Felsenstein, <a href="/A005373/a005373.pdf">The number of evolutionary trees</a>, Systematic Zoology, 27 (1978), 27-33. (Annotated scanned copy)

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/PerfectMatchings">Perfect matchings</a>

%H A. Genitrini, B. Gittenberger, M. Kauers and M. Wallner, <a href="https://arxiv.org/abs/1703.10031">Asymptotic Enumeration of Compacted Binary Trees</a>, arXiv:1703.10031 [math.CO], 2017

%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

%H S. Goodenough, C. Lavault, <a href="http://arxiv.org/abs/1404.1894">On subsets of Riordan subgroups and Heisenberg--Weyl algebra</a>, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.

%H S. Goodenough, C. Lavault, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p16">Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups</a>, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16.

%H W. S. Gray and M. Thitsa, <a href="http://dx.doi.org/10.1109/SSST.2013.6524939">System Interconnections and Combinatorial Integer Sequences</a>, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-Mar 11 2013.

%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=23">Encyclopedia of Combinatorial Structures 23</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=106">Encyclopedia of Combinatorial Structures 106</a>

%H M. Kauers and S.-L. Ko, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.01.084">Problem 11545</a>, Amer. Math. Monthly, 118 (2011), p. 84.

%H A. Khruzin, <a href="http://arXiv.org/abs/math.CO/0008209">Enumeration of chord diagrams</a>, arXiv:math/0008209 [math.CO], 2000.

%H M. Klazar, <a href="http://dx.doi.org/10.1006/eujc.1995.0095">Twelve countings with rooted plane trees</a>, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.

%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H F. Larrion, M. A. Pizana, and R. Villarroel-Flores, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.047">The clique operator on matching and chessboard graphs</a> Discrete Math. 309 (2009), no. 1, 85-93.

%H E. Lucas, <a href="/A000899/a000899.pdf">Theorie des nombres</a> (annotated scans of a few selected pages)

%H Robert J. Marsh and Paul Martin, <a href="http://www.emis.de/journals/JACO/Volume33_3/7524r60365n3754j.html">Tiling bijections between paths and Brauer diagrams</a>, Journal of Algebraic Combinatorics, Vol 33, No 3 (2011), p 427-453.

%H B. E. Meserve, <a href="http://www.jstor.org/stable/2306136">Double Factorials</a>, American Mathematical Monthly, 55 (1948), 425-426.

%H T. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1945-08486-9">The hypersurface cross ratio</a>, Bull. Amer. Math. Soc., 51 (1945), 976-984.

%H T. S. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1948-09002-4 ">Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products</a>, Bull. Amer. Math. Soc., 54 (1948), 352-360.

%H F. Murtagh, <a href="http://dx.doi.org/10.1016/0166-218X(84)90066-0">Counting dendrograms: a survey</a>, Discrete Applied Mathematics, 7 (1984), 191-199.

%H G. Nordh, <a href="http://arXiv.org/abs/math.NT/0506155">Perfect Skolem sequences</a>, arXiv:math/0506155 [math.CO], 2005.

%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

%H R. Ondrejka, <a href="http://dx.doi.org/10.1090/S0025-5718-70-99856-X">Tables of double factorials</a>, Math. Comp., 24 (1970), 231.

%H L. Pachter and B. Sturmfels, <a href="http://arXiv.org/abs/math.ST/0409132">The mathematics of phylogenomics</a>, arXiv:math/0409132 [math.ST], 2004-2005.

%H K. Phillips, <a href="http://www.jstatsoft.org/v33/c01/paper">R functions to symbolically compute the central moments of the multivariate normal distribution</a>, Journal of Statistical Software, Feb 2010.

%H R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.

%H Helmut Prodinger, <a href="http://www.combinatorics.org/Volume_3/volume3.html#R29">Descendants in heap ordered trees or a triumph of computer algebra </a>

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q541.htm">Question 541</a>, J. Ind. Math. Soc.

%H M. D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Int. Seq. 13 (2010), 10.6.7, (6.27)

%H E. Schröder, <a href="http://resolver.sub.uni-goettingen.de/purl?PPN599415665_0015">Vier combinatorische Probleme</a>, Z. f. Math. Phys., 15 (1870), 361-376.

%H E. Schröder, <a href="/A000108/a000108_9.pdf">Vier combinatorische Probleme</a>, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy]

%H Y. S. Song, <a href="http://dx.doi.org/10.1007/s00026-003-0192-0">On the combinatorics of rooted binary phylogenetic trees</a>, Annals of Combinatorics, 7, 2003, 365-379. See Lemma 2.1. - _N. J. A. Sloane_, Aug 22 2014

%H Andrew Vince and Miklos Bona, <a href="http://arxiv.org/abs/1204.3842">The Number of Ways to Assemble a Graph</a>, arXiv preprint arXiv:1204.3842 [math.CO], 2012.

%H Michael Wallner, <a href="https://arxiv.org/abs/1703.10031">A bijection of plane increasing trees with relaxed binary trees of right height at most one</a>, arXiv:1706.07163 [math.CO], 2017

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erf.html">Erf</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NormalDistributionFunction.html">Normal Distribution Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pfaffian">Pfaffian</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F E.g.f.: 1 / sqrt(1 - 2*x).

%F a(n) = a(n-1)*(2*n-1) = (2*n)!/(n!*2^n) = A010050(n)/A000165(n).

%F a(n) ~ sqrt(2) * 2^n * (n/e)^n.

%F Rational part of numerator of Gamma(n+1/2): a(n) * sqrt(Pi) / 2^n = Gamma(n+1/2). - Yuriy Brun, Ewa Dominowska (brun(AT)mit.edu), May 12 2001

%F With interpolated zeros, the sequence has e.g.f. exp(x^2/2). - _Paul Barry_, Jun 27 2003

%F The Ramanujan polynomial psi(n+1, n) has value a(n). - _Ralf Stephan_, Apr 16 2004

%F a(n) = Sum_{k=0..n} (-2)^(n-k)*A048994(n, k). - _Philippe Deléham_, Oct 29 2005

%F Log(1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + ...) = x + 5/2*x^2 + 37/3*x^3 + 353/4*x^4 + 4081/5*x^5 + 55205/6*x^6 + ..., where [1, 5, 37, 353, 4081, 55205, ...] = A004208. - _Philippe Deléham_, Jun 20 2006

%F 1/3 + 2/15 + 3/105 + ... = 1/2. [Jolley eq. 216]

%F Sum_{j=1..n} j/a(j+1) = (1 - 1/a(n+1))/2. [Jolley eq. 216]

%F 1/1 + 1/3 + 2/15 + 6/105 + 24/945 + ... = Pi/2. - _Gary W. Adamson_, Dec 21 2006

%F a(n) = (1/sqrt(2*Pi))*Integral_{x>=0} x^n*exp(-x/2)/sqrt(x). - _Paul Barry_, Jan 28 2008

%F a(n) = A006882(2n-1). - _R. J. Mathar_, Jul 04 2009

%F G.f.: 1/(1-x-2x^2/(1-5x-12x^2/(1-9x-30x^2/(1-13x-56x^2/(1- ... (continued fraction). - _Paul Barry_, Sep 18 2009

%F a(n) = (-1)^n*subs({log(e)=1,x=0},coeff(simplify(series(e^(x*t-t^2/2),t,2*n+1)),t^(2*n))*(2*n)!). - _Leonid Bedratyuk_, Oct 31 2009

%F a(n) = 2^n*gamma(n+1/2)/gamma(1/2). - _Jaume Oliver Lafont_, Nov 09 2009

%F G.f.: 1/(1-x/(1-2x/(1-3x/(1-4x/(1-5x/(1- ...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009

%F The g.f. of a(n+1) is 1/(1-3x/(1-2x/(1-5x/(1-4x/(1-7x/(1-6x/(1-.... (continued fraction). - _Paul Barry_, Dec 04 2009

%F a(n) = Sum_{i=1..n} binomial(n,i)*a(i-1)*a(n-i). - _Vladimir Shevelev_, Sep 30 2010

%F E.g.f.: A(x) = 1 - sqrt(1-2*x) satisfies the differential equation A'(x) - A'(x)*A(x) - 1 = 0. - _Vladimir Kruchinin_, Jan 17 2011

%F a(n) = A123023(2*n + 1). - _Michael Somos_, Jul 24 2011

%F a(n) = (1/2)*Sum_{i=1..n} binomial(n+1,i)*a(i-1)*a(n-i). See link above. - _Dennis P. Walsh_, Dec 02 2011

%F a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,n+k)*Stirling_1(n+k,k) [Kauers and Ko].

%F a(n) = A035342(n, 1), n >= 1 (first column of triangle).

%F a(n) = A001497(n, 0) = A001498(n, n), first column, resp. main diagonal, of Bessel triangle.

%F From _Gary W. Adamson_, Jul 19 2011: (Start)

%F a(n) = upper left term of M^n and sum of top row terms of M^(n-1), where M = a variant of the (1,2) Pascal triangle (Cf. A029635) as the following production matrix:

%F 1, 2, 0, 0, 0, ...

%F 1, 3, 2, 0, 0, ...

%F 1, 4, 5, 2, 0, ...

%F 1, 5, 9, 7, 2, ...

%F ...

%F For example, a(3) = 15 is the left term in top row of M^3: (15, 46, 36, 8) and a(4) = 105 = (15 + 46 + 36 + 8).

%F (End)

%F G.f.: A(x)=1+x/(W(0)-x); W(k)=1+x+x*2k-x*(2k+3)/W(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 17 2011

%F a(n) = Sum_{i=1..n} binomial(n,i-1)*a(i-1)*a(n-i). - _Dennis P. Walsh_, Dec 02 2011

%F a(n) = A009445(n) / A014481(n). - _Reinhard Zumkeller_, Dec 03 2011

%F a(n) = (-1)^n*Sum_{k=0..n} 2^(n-k)*s(n+1,k+1), where s(n,k) are the Stirling numbers of the first kind, A048994. - _Mircea Merca_, May 03 2012

%F a(n) = (2*n)_4! = Gauss_factorial(2*n,4) = Product_{j=1..2*n, gcd(j,4)=1} j. - _Peter Luschny_, Oct 01 2012

%F G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k-1)/(1 - x*(2*k+2)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 19 2013

%F G.f.: 1 + x/Q(0), where Q(k)= 1 + (2*k-1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 01 2013

%F G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 31 2013

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013

%F G.f.: G(0), where G(k)= 1 + 2*x*(4*k+1)/(4*k+2 - 2*x*(2*k+1)*(4*k+3)/(x*(4*k+3) + 2*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 22 2013

%F a(n) = (2n-3)*a(n-2) + (2n-2)*a(n-1), n>1. - _Ivan N. Ianakiev_, Jul 08 2013

%F G.f.: G(0), where G(k)= 1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 04 2013

%F a(n) = 2*a(n-1) + (2n-3)^2*a(n-2), a(0) = a(1) = 1. - _Philippe Deléham_, Oct 27 2013

%F G.f. of reciprocals: Sum_{n>=0} x^n/a(n) = 1F1(1;1/2;x/2), confluent hypergeometric Function. - _R. J. Mathar_, Jul 25 2014

%F 0 = a(n)*(+2*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) for all n in Z. - _Michael Somos_, Sep 18 2014

%F a(n) = (-1)^n / a(-n) = 2*a(n-1) + a(n-1)^2 / a(n-2) for all n in Z. - _Michael Somos_, Sep 18 2014

%F From _Peter Bala_, Feb 18 2015: (Start)

%F Recurrence equation: a(n) = (3*n - 2)*a(n-1) - (n - 1)*(2*n - 3)*a(n-2) with a(1) = 1 and a(2) = 3.

%F The sequence b(n) = A087547(n), beginning [1, 4, 52, 608, 12624, ... ], satisfies the same second-order recurrence equation. This leads to the generalized continued fraction expansion lim_{n -> inf} b(n)/a(n) = Pi/2 = 1 + 1/(3 - 6/(7 - 15/(10 - ... - n*(2*n - 1)/((3*n + 1) - ... )))). (End)

%F E.g.f of the sequence whose n-th element (n = 1,2,...) equals a(n-1) is 1-sqrt(1-2*x). - _Stanislav Sykora_, Jan 06 2017

%F Sum_{n>=1} a(n)/(2*n-1)! = exp(1/2). - _Daniel Suteu_, Feb 06 2017

%F a(n) = A028338(n, 0), n >= 0. - _Wolfdieter Lang_, May 27 2017

%F a(n) = (Product_{k=0..n-2} binomial(2*(n-k),2))/n!. - _Stefano Spezia_, Nov 13 2018

%e a(3) = 1*3*5 = 15.

%e From _Joerg Arndt_, Sep 10 2013: (Start)

%e There are a(3)=15 involutions of 6 elements without fixed points:

%e #: permutation transpositions

%e 01: [ 1 0 3 2 5 4 ] (0, 1) (2, 3) (4, 5)

%e 02: [ 1 0 4 5 2 3 ] (0, 1) (2, 4) (3, 5)

%e 03: [ 1 0 5 4 3 2 ] (0, 1) (2, 5) (3, 4)

%e 04: [ 2 3 0 1 5 4 ] (0, 2) (1, 3) (4, 5)

%e 05: [ 2 4 0 5 1 3 ] (0, 2) (1, 4) (3, 5)

%e 06: [ 2 5 0 4 3 1 ] (0, 2) (1, 5) (3, 4)

%e 07: [ 3 2 1 0 5 4 ] (0, 3) (1, 2) (4, 5)

%e 08: [ 3 4 5 0 1 2 ] (0, 3) (1, 4) (2, 5)

%e 09: [ 3 5 4 0 2 1 ] (0, 3) (1, 5) (2, 4)

%e 10: [ 4 2 1 5 0 3 ] (0, 4) (1, 2) (3, 5)

%e 11: [ 4 3 5 1 0 2 ] (0, 4) (1, 3) (2, 5)

%e 12: [ 4 5 3 2 0 1 ] (0, 4) (1, 5) (2, 3)

%e 13: [ 5 2 1 4 3 0 ] (0, 5) (1, 2) (3, 4)

%e 14: [ 5 3 4 1 2 0 ] (0, 5) (1, 3) (2, 4)

%e 15: [ 5 4 3 2 1 0 ] (0, 5) (1, 4) (2, 3)

%e (End)

%e G.f. = 1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + 135135*x^7 + ...

%p f := n->(2*n)!/(n!*2^n);

%p A001147 := proc(n) doublefactorial(2*n-1); end: # _R. J. Mathar_, Jul 04 2009

%p A001147 := n -> 2^n*pochhammer(1/2, n); # _Peter Luschny_, Aug 09 2009

%p G(x):=(1-2*x)^(-1/2): 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..19); # _Zerinvary Lajos_, Apr 03 2009; aligned with offset by _Johannes W. Meijer_, Aug 11 2009

%p series(hypergeom([1,1/2],[],2*x),x=0,20); # _Mark van Hoeij_, Apr 07 2013

%t Table[(2 n - 1)!!, {n, 0, 19}] (* _Robert G. Wilson v_, Oct 12 2005 *)

%t a[ n_] := 2^n Gamma[n + 1/2] / Gamma[1/2]; (* _Michael Somos_, Sep 18 2014 *)

%t Join[{1}, Range[1, 41, 2]!!] (* _Harvey P. Dale_, Jan 28 2017 *)

%t a[ n_] := If[ n < 0, (-1)^n / a[-n], SeriesCoefficient[ Product[1 - (1 - x)^(2 k - 1), {k, n}], {x, 0, n}]]; (* _Michael Somos_, Jun 27 2017 *)

%t (2 Range[0, 20] - 1)!! (* _Eric W. Weisstein_, Jul 22 2017 *)

%o (PARI) {a(n) = if( n<0, (-1)^n / a(-n), (2*n)! / n! / 2^n)}; /* _Michael Somos_, Sep 18 2014 */

%o (PARI) x='x+O('x^33); Vec(serlaplace((1-2*x)^(-1/2))) \\ _Joerg Arndt_, Apr 24 2011

%o (MAGMA) A001147:=func< n | n eq 0 select 1 else &*[ k: k in [1..2*n-1 by 2] ] >; [ A001147(n): n in [0..20] ]; // _Klaus Brockhaus_, Jun 22 2011

%o (MAGMA) I:=[1,3]; [1] cat [n le 2 select I[n] else (3*n-2)*Self(n-1)-(n-1)*(2*n-3)*Self(n-2): n in [1..25] ]; // _Vincenzo Librandi_, Feb 19 2015

%o a001147 n = product [1, 3 .. 2 * n - 1]

%o a001147_list = 1 : zipWith (*) [1, 3 ..] a001147_list

%o -- _Reinhard Zumkeller_, Feb 15 2015, Dec 03 2011

%o (Sage) [rising_factorial(n+1,n)/2^n for n in (0..15)] # _Peter Luschny_, Jun 26 2012

%o (Python)

%o from sympy import factorial2

%o def a(n): return factorial2(2*n - 1)

%o print map(a, xrange(101)) # _Indranil Ghosh_, Jul 22 2017

%o (GAP) A001147 := function(n) local i, s, t; t := 1; i := 0; Print(t, ", "); for i in [1 .. n] do t := t*(2*i-1); Print(t, ", "); od; end; A001147(100); # _Stefano Spezia_, Nov 13 2018

%Y Cf. A000085, A006882, A000165 ((2n)!!), A001818, A009445, A039683, A102992, A001190 (no labels), A000680, A132101.

%Y Cf. A086677; A055142 (for this sequence, |a(n+1)| + 1 is the number of distinct products which can be formed using commutative, nonassociative multiplication and a nonempty subset of n given variables).

%Y Constant terms of polynomials in A098503. First row of array A099020.

%Y Cf. A079267, A000698, A029635, A161198, A076795, A123023, A161124, A051125, A181983, A099174, A087547, A028338 (first column).

%Y Subsequence of A248652.

%Y Cf. A082161 (relaxed compacted binary trees of unbounded right height).

%K nonn,easy,nice,core,changed

%O 0,3

%A _N. J. A. Sloane_

%E Removed erroneous comments: neither the number of n X n binary matrices A such that A^2 = 0 nor the number of simple directed graphs on n vertices with no directed path of length two are counted by this sequence (for n = 3, both are 13). - _Dan Drake_, Jun 02 2009

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Last modified November 14 21:12 EST 2018. Contains 317217 sequences. (Running on oeis4.)