

A001147


Double factorial of odd numbers: a(n) = (2*n1)!! = 1*3*5*...*(2*n1).
(Formerly M3002 N1217)


545



1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575, 316234143225, 7905853580625, 213458046676875, 6190283353629375, 191898783962510625, 6332659870762850625, 221643095476699771875, 8200794532637891559375
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OFFSET

0,3


COMMENTS

The solution to Schröder's third problem.
Number of fixedpointfree involutions in symmetric group S_{2n} (cf. A000085).
a(n+2) is the number of full Steiner topologies on n points with n2 Steiner points.
a(n) is also the number of perfect matchings in the complete graph K(2n).  Ola Veshta (olaveshta(AT)mydeja.com), Mar 25 2001
Number of ways to choose n disjoint pairs of items from 2*n items.  Ron Zeno (rzeno(AT)hotmail.com), Feb 06 2002
Number of ways to choose n1 disjoint pairs of items from 2*n1 items (one item remains unpaired).  Bartosz Zoltak, Oct 16 2012
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)mydeja.com), Apr 21 2001
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).
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
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 secondorder Eulerian number (A008517).  Ross La Haye, Feb 13 2005
Integral representation as nth 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
a(n) is the number of binary total partitions of n+1 (each nonsingleton 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
a(n) is the Pfaffian of the skewsymmetric 2n X 2n matrix whose (i,j) entry is i for i<j.  David Callan, Sep 25 2006
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
Number of perfect multi Skolemtype sequences of order n.  Emeric Deutsch, Nov 24 2006
a(n) = total weight of all Dyck npaths (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 3paths 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
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
From Tom Copeland, Nov 13 2007, clarified in first and extended in second paragraph, Jun 12 2021: (Start)
a(n) has the e.g.f. (12x)^(1/2) = 1 + x + 3*x^2/2! + ..., whose reciprocal is (12x)^(1/2) = 1  x  x^2/2!  3*x^3/3!  ... = b(0)  b(1)*x  b(2)*x^2/2!  ... with b(0) = 1 and b(n+1) = a(n) otherwise. By the formalism of A133314, Sum_{k=0..n} binomial(n,k)*b(k)*a(nk) = 0^n where 0^0 := 1. In this sense, the sequence a(n) is essentially selfinverse. See A132382 for an extension of this result. See A094638 for interpretations.
This sequence aerated has the e.g.f. e^(t^2/2) = 1 + t^2/2! + 3*t^4/4! + ... = c(0) + c(1)*t + c(2)*t^2/2! + ... and the reciprocal e^(t^2/2); therefore, Sum_{k=0..n} cos(Pi k/2)*binomial(n,k)*c(k)*c(nk) = 0^n; i.e., the aerated sequence is essentially selfinverse. Consequently, Sum_{k=0..n} (1)^k*binomial(2n,2k)*a(k)*a(nk) = 0^n. (End)
From Ross Drewe, Mar 16 2008: (Start)
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.
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.
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] }.
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]}, ....}
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)
A139541(n) = a(n) * a(2*n).  Reinhard Zumkeller, Apr 25 2008
a(n+1) = Sum_{j=0..n} A074060(n,j) * 2^j.  Tom Copeland, Sep 01 2008
From Emeric Deutsch, Jun 05 2009: (Start)
a(n) is the number of adjacent transpositions in all fixedpointfree 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.
a(n) = Sum_{k>=0} k*A079267(n,k).
(End)
Hankel transform is A137592.  Paul Barry, Sep 18 2009
(1, 3, 15, 105, ...) = INVERT transform of A000698 starting (1, 2, 10, 74, ...).  Gary W. Adamson, Oct 21 2009
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*tt^2/2) = Sum_{i>=0} H(i,x)*t^i/i!.  Leonid Bedratyuk, Oct 31 2009
The Hankel transform of a(n+1) is A168467.  Paul Barry, Dec 04 2009
Partial products of odd numbers.  JuriStepan Gerasimov, Oct 17 2010
See A094638 for connections to differential operators.  Tom Copeland, Sep 20 2011
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
a(n) is the leading coefficient of the Bessel polynomial y_n(x) (cf. A001498).  Leonid Bedratyuk, Jun 01 2012
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
a(n) is also the numerator of the mean value from 0 to Pi/2 of sin(x)^(2n).  JeanFrançois Alcover, Jun 13 2013
a(n) is the size of the Brauer monoid on 2n points (see A227545).  James Mitchell, Jul 28 2013
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) ~ n1/2 (for example, large KendellMann numbers, see A000140 or A181609, as n > infinity).  Mikhail Gaichenkov, Jan 14 2014
a(n) = the number of uppertriangular matrix representations required for the symbolic representation of a first order central moment of the multivariate normal distribution of dimension 2(n1), i.e., E[X_1*X_2...*X_(2n2)mu=0, Sigma]. See vignette for symmoments R package on CRAN and Phillips reference below.  Kem Phillips, Aug 10 2014
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
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
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 postorder 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 postorder traversal. The right height is the maximal number of rightedges (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
Also the number of distinct adjacency matrices in the nladder rung graph.  Eric W. Weisstein, Jul 22 2017
From Christopher J. Smyth, Jan 26 2018: (Start)
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.)
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 2partition of T. This also follows from Mitch Harris' comment above, since the leaf labels determine the labels of the other vertices of the tree.
(End)
a(n) is the nth moment of the chisquared distribution with one degree of freedom (equivalent to Coleman's Apr 06 2004 comment).  Bryan R. Gillespie, Mar 07 2021
Let b(n) = 0 for n odd and b(2k) = a(k); i.e., let the sequence b(n) be an aerated version of this entry. After expanding the differential operator (x + D)^n and normal ordering the resulting terms, the integer coefficient of the term x^k D^m is n! b(nkm) / [(nkm)! k! m!] with 0 <= k,m <= n and (k+m) <= n. E.g., (x+D)^2 = x^2 + 2xD + D^2 + 1 with D = d/dx. The result generalizes to the raising (R) and lowering (L) operators of any Sheffer polynomial sequence by replacing x by R and D by L and follows from the disentangling relation e^{t(L+R)} = e^{t^2/2} e^{tR} e^{tL}. Consequently, these are also the coefficients of the reordered 2^n permutations of the binary symbols L and R under the condition LR = RL + 1. E.g., (L+R)^2 = LL + LR + RL + RR = LL + 2RL + RR + 1. (Cf. A344678.)  Tom Copeland, May 25 2021
From Tom Copeland, Jun 14 2021: (Start)
Lando and Zvonkin present several scenarios in which the double factorials occur in their role of enumerating perfect matchings (pairings) and as the nonzero moments of the Gaussian e^(x^2/2!).
Speyer and Sturmfels (p. 6) state that the number of facets of the abstract simplicial complex known as the tropical Grassmannian G'''(2,n), the space of phylogenetic T_n trees (see A134991), or Whitehouse complex is a shifted double factorial.
These are also the unsigned coefficients of the x[2]^m terms in the partition polynomials of A134685 for compositional inversion of e.g.f.s, a refinement of A134991.
a(n)*2^n = A001813(n) and A001813(n)/(n+1)! = A000108(n), the Catalan numbers, the unsigned coefficients of the x[2]^m terms in the partition polynomials A133437 for compositional inversion of o.g.f.s, a refinement of A033282, A126216, and A086810. Then the double factorials inherit a multitude of analytic and combinatoric interpretations from those of the Catalan numbers, associahedra, and the noncrossing partitions of A134264 with the Catalan numbers as unsignedrow sums. (End)


REFERENCES

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).
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 317.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228, #19.
Hoel, Port and Stone, Introduction to Probability Theory, Section 7.3.
F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, NorthHolland, 1992, see p. 14.
C. Itzykson and J.B. Zuber, Quantum Field Theory, McGrawHill, 1980, pages 466467.
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 48.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.6 and also p. 178.
R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, SpringerVerlag, New York, 1999, p. 73.


LINKS

Stefano Spezia, Table of n, a(n) for n = 0..400 (first 102 terms from T. D. Noe)
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].
Jonathan Burns, Assembly Graph Words  Single Transverse Component (Counts).
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double, Hyper, Suband Superfactorials, The American Mathematical Monthly 122.5 (2015): 433443.
D. Arques and J.F. Beraud, Rooted maps on orientable surfaces, Riccati's equation and continued fractions, Discrete Math., 215 (2000), 112.
Fatemeh Bagherzadeh, M. Bremner, and S. Madariaga, Jordan Trialgebras and PostJordan Algebras, arXiv preprint arXiv:1611.01214 [math.RA], 2016.
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, On the fMatrices of Pascallike Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
Paul Barry and A. Hennessy, The EulerSeidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2.
Natasha Blitvić and Einar Steingrímsson, Permutations, moments, measures, arXiv:2001.00280 [math.CO], 2020.
O. Bodini, M. Dien, X. Fontaine, A. Genitrini, and H. K. Hwang, Increasing Diamonds, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 1115, 2016, Proceedings Pages pp. 207219 2016 DOI 10.1007/9783662495292_16; Lecture Notes in Computer Science Series Volume 9644.
H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, FourRegular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011.
David Callan, A combinatorial survey of identities for the double factorial, arXiv:0906.1317 [math.CO], 2009.
Peter J. Cameron, Some treelike objects Quart. J. Math. Oxford Ser. 38 (1987), 155183. MR0891613 (89a:05009). See p. 155.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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P. Codara, O. M. D'Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
Thierry DanaPicard, Sequences of Definite Integrals, Factorials and Double Factorials, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.6.
Filippo Disanto and Thomas Wiehe, Some combinatorial problems on binary rooted trees occurring in population genetics, arXiv preprint arXiv:1112.1295 [math.CO], 2011.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 5.
J. Felsenstein, The number of evolutionary trees, Systematic Zoology, 27 (1978), 2733. (Annotated scanned copy)
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A. Genitrini, B. Gittenberger, M. Kauers and M. Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and HeisenbergWeyl algebra, arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16.
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M. Kauers and S.L. Ko, Problem 11545, Amer. Math. Monthly, 118 (2011), p. 84.
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J.C. Novelli and J.Y. Thibon, Hopf Algebras of mpermutations,(m+1)ary trees, and mparking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
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D. Speyer and B. Sturmfels, The tropical Grassmannian, arXiv:0304218 [math.AG], 2003.
Neriman Tokcan, Jonathan Gryak, Kayvan Najarian, and Harm Derksen, Algebraic Methods for Tensor Data, arXiv:2005.12988 [math.RT], 2020.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Double Factorial
Eric Weisstein's World of Mathematics, Erf
Eric Weisstein's World of Mathematics, Ladder Rung Graph
Eric Weisstein's World of Mathematics, Normal Distribution Function
Wikipedia, Pfaffian
Wikipedia, Hermite polynomials
Index to divisibility sequences
Index entries for related partitioncounting sequences
Index entries for sequences related to factorial numbers
Index entries for sequences related to parenthesizing
Index entries for "core" sequences


FORMULA

E.g.f.: 1 / sqrt(1  2*x).
Dfinite with recurrence: a(n) = a(n1)*(2*n1) = (2*n)!/(n!*2^n) = A010050(n)/A000165(n).
a(n) ~ sqrt(2) * 2^n * (n/e)^n.
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
With interpolated zeros, the sequence has e.g.f. exp(x^2/2).  Paul Barry, Jun 27 2003
The Ramanujan polynomial psi(n+1, n) has value a(n).  Ralf Stephan, Apr 16 2004
a(n) = Sum_{k=0..n} (2)^(nk)*A048994(n, k).  Philippe Deléham, Oct 29 2005
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
1/3 + 2/15 + 3/105 + ... = 1/2. [Jolley eq. 216]
Sum_{j=1..n} j/a(j+1) = (1  1/a(n+1))/2. [Jolley eq. 216]
1/1 + 1/3 + 2/15 + 6/105 + 24/945 + ... = Pi/2.  Gary W. Adamson, Dec 21 2006
a(n) = (1/sqrt(2*Pi))*Integral_{x>=0} x^n*exp(x/2)/sqrt(x).  Paul Barry, Jan 28 2008
a(n) = A006882(2n1).  R. J. Mathar, Jul 04 2009
G.f.: 1/(1x2x^2/(15x12x^2/(19x30x^2/(113x56x^2/(1 ... (continued fraction).  Paul Barry, Sep 18 2009
a(n) = (1)^n*subs({log(e)=1,x=0},coeff(simplify(series(e^(x*tt^2/2),t,2*n+1)),t^(2*n))*(2*n)!).  Leonid Bedratyuk, Oct 31 2009
a(n) = 2^n*gamma(n+1/2)/gamma(1/2).  Jaume Oliver Lafont, Nov 09 2009
G.f.: 1/(1x/(12x/(13x/(14x/(15x/(1 ...(continued fraction).  Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
The g.f. of a(n+1) is 1/(13x/(12x/(15x/(14x/(17x/(16x/(1.... (continued fraction).  Paul Barry, Dec 04 2009
a(n) = Sum_{i=1..n} binomial(n,i)*a(i1)*a(ni).  Vladimir Shevelev, Sep 30 2010
E.g.f.: A(x) = 1  sqrt(12*x) satisfies the differential equation A'(x)  A'(x)*A(x)  1 = 0.  Vladimir Kruchinin, Jan 17 2011
a(n) = A123023(2*n + 1).  Michael Somos, Jul 24 2011
a(n) = (1/2)*Sum_{i=1..n} binomial(n+1,i)*a(i1)*a(ni). See link above.  Dennis P. Walsh, Dec 02 2011
a(n) = Sum_{k=0..n} (1)^k*binomial(2*n,n+k)*Stirling_1(n+k,k) [Kauers and Ko].
a(n) = A035342(n, 1), n >= 1 (first column of triangle).
a(n) = A001497(n, 0) = A001498(n, n), first column, resp. main diagonal, of Bessel triangle.
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n and sum of top row terms of M^(n1), where M = a variant of the (1,2) Pascal triangle (Cf. A029635) as the following production matrix:
1, 2, 0, 0, 0, ...
1, 3, 2, 0, 0, ...
1, 4, 5, 2, 0, ...
1, 5, 9, 7, 2, ...
...
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).
(End)
G.f.: A(x) = 1 + x/(W(0)  x); W(k) = 1 + x + x*2*k  x*(2*k + 3)/W(k+1); (continued fraction).  Sergei N. Gladkovskii, Nov 17 2011
a(n) = Sum_{i=1..n} binomial(n,i1)*a(i1)*a(ni).  Dennis P. Walsh, Dec 02 2011
a(n) = A009445(n) / A014481(n).  Reinhard Zumkeller, Dec 03 2011
a(n) = (1)^n*Sum_{k=0..n} 2^(nk)*s(n+1,k+1), where s(n,k) are the Stirling numbers of the first kind, A048994.  Mircea Merca, May 03 2012
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
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
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
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
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
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
a(n) = (2*n  3)*a(n2) + (2*n  2)*a(n1), n > 1.  Ivan N. Ianakiev, Jul 08 2013
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
a(n) = 2*a(n1) + (2n3)^2*a(n2), a(0) = a(1) = 1.  Philippe Deléham, Oct 27 2013
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
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
a(n) = (1)^n / a(n) = 2*a(n1) + a(n1)^2 / a(n2) for all n in Z.  Michael Somos, Sep 18 2014
From Peter Bala, Feb 18 2015: (Start)
Recurrence equation: a(n) = (3*n  2)*a(n1)  (n  1)*(2*n  3)*a(n2) with a(1) = 1 and a(2) = 3.
The sequence b(n) = A087547(n), beginning [1, 4, 52, 608, 12624, ... ], satisfies the same secondorder 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)
E.g.f of the sequence whose nth element (n = 1,2,...) equals a(n1) is 1sqrt(12*x).  Stanislav Sykora, Jan 06 2017
Sum_{n >= 1} a(n)/(2*n1)! = exp(1/2).  Daniel Suteu, Feb 06 2017
a(n) = A028338(n, 0), n >= 0.  Wolfdieter Lang, May 27 2017
a(n) = (Product_{k=0..n2} binomial(2*(nk),2))/n!.  Stefano Spezia, Nov 13 2018
a(n) = Sum_{i=0..n1} Sum_{j=0..ni1} C(n1,i)*C(ni1,j)*a(i)*a(j)*a(nij1), a(0)=1,  Vladimir Kruchinin, May 06 2020
From Amiram Eldar, Jun 29 2020: (Start)
Sum_{n>=1} 1/a(n) = sqrt(e*Pi/2)*erf(1/sqrt(2)), where erf is the error function.
Sum_{n>=1} (1)^(n+1)/a(n) = sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. (End)


EXAMPLE

a(3) = 1*3*5 = 15.
From Joerg Arndt, Sep 10 2013: (Start)
There are a(3)=15 involutions of 6 elements without fixed points:
#: permutation transpositions
01: [ 1 0 3 2 5 4 ] (0, 1) (2, 3) (4, 5)
02: [ 1 0 4 5 2 3 ] (0, 1) (2, 4) (3, 5)
03: [ 1 0 5 4 3 2 ] (0, 1) (2, 5) (3, 4)
04: [ 2 3 0 1 5 4 ] (0, 2) (1, 3) (4, 5)
05: [ 2 4 0 5 1 3 ] (0, 2) (1, 4) (3, 5)
06: [ 2 5 0 4 3 1 ] (0, 2) (1, 5) (3, 4)
07: [ 3 2 1 0 5 4 ] (0, 3) (1, 2) (4, 5)
08: [ 3 4 5 0 1 2 ] (0, 3) (1, 4) (2, 5)
09: [ 3 5 4 0 2 1 ] (0, 3) (1, 5) (2, 4)
10: [ 4 2 1 5 0 3 ] (0, 4) (1, 2) (3, 5)
11: [ 4 3 5 1 0 2 ] (0, 4) (1, 3) (2, 5)
12: [ 4 5 3 2 0 1 ] (0, 4) (1, 5) (2, 3)
13: [ 5 2 1 4 3 0 ] (0, 5) (1, 2) (3, 4)
14: [ 5 3 4 1 2 0 ] (0, 5) (1, 3) (2, 4)
15: [ 5 4 3 2 1 0 ] (0, 5) (1, 4) (2, 3)
(End)
G.f. = 1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + 135135*x^7 + ...


MAPLE

f := n>(2*n)!/(n!*2^n);
A001147 := proc(n) doublefactorial(2*n1); end: # R. J. Mathar, Jul 04 2009
A001147 := n > 2^n*pochhammer(1/2, n); # Peter Luschny, Aug 09 2009
G(x):=(12*x)^(1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n1], 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
series(hypergeom([1, 1/2], [], 2*x), x=0, 20); # Mark van Hoeij, Apr 07 2013


MATHEMATICA

Table[(2 n  1)!!, {n, 0, 19}] (* Robert G. Wilson v, Oct 12 2005 *)
a[ n_] := 2^n Gamma[n + 1/2] / Gamma[1/2]; (* Michael Somos, Sep 18 2014 *)
Join[{1}, Range[1, 41, 2]!!] (* Harvey P. Dale, Jan 28 2017 *)
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 *)
(2 Range[0, 20]  1)!! (* Eric W. Weisstein, Jul 22 2017 *)


PROG

(PARI) {a(n) = if( n<0, (1)^n / a(n), (2*n)! / n! / 2^n)}; /* Michael Somos, Sep 18 2014 */
(PARI) x='x+O('x^33); Vec(serlaplace((12*x)^(1/2))) \\ Joerg Arndt, Apr 24 2011
(MAGMA) A001147:=func< n  n eq 0 select 1 else &*[ k: k in [1..2*n1 by 2] ] >; [ A001147(n): n in [0..20] ]; // Klaus Brockhaus, Jun 22 2011
(MAGMA) I:=[1, 3]; [1] cat [n le 2 select I[n] else (3*n2)*Self(n1)(n1)*(2*n3)*Self(n2): n in [1..25] ]; // Vincenzo Librandi, Feb 19 2015
(Haskell)
a001147 n = product [1, 3 .. 2 * n  1]
a001147_list = 1 : zipWith (*) [1, 3 ..] a001147_list
 Reinhard Zumkeller, Feb 15 2015, Dec 03 2011
(Sage) [rising_factorial(n+1, n)/2^n for n in (0..15)] # Peter Luschny, Jun 26 2012
(Python)
from sympy import factorial2
def a(n): return factorial2(2 * n  1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 22 2017
(GAP) A001147 := function(n) local i, s, t; t := 1; i := 0; Print(t, ", "); for i in [1 .. n] do t := t*(2*i1); Print(t, ", "); od; end; A001147(100); # Stefano Spezia, Nov 13 2018
(Maxima)
a(n):=if n=0 then 1 else sum(sum(binomial(n1, i)*binomial(ni1, j)*a(i)*a(j)*a(nij1), j, 0, ni1), i, 0, n1); /* Vladimir Kruchinin, May 06 2020 */


CROSSREFS

Cf. A000085, A006882, A000165 ((2n)!!), A001818, A009445, A039683, A102992, A001190 (no labels), A000680, A132101.
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).
Constant terms of polynomials in A098503. First row of array A099020.
Cf. A079267, A000698, A029635, A161198, A076795, A123023, A161124, A051125, A181983, A099174, A087547, A028338 (first column).
Subsequence of A248652.
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A053871 (binomial transform).
Cf. A000108, A001813, A033282, A060540, A086810, A094638, A126216, A133437, A134264, A134685, A134991, A344678.
Sequence in context: A294000 A294001 A294002 * A330797 A000268 A207818
Adjacent sequences: A001144 A001145 A001146 * A001148 A001149 A001150


KEYWORD

nonn,easy,nice,core


AUTHOR

N. J. A. Sloane


EXTENSIONS

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


STATUS

approved



