%I A007317 M1480
%S A007317 1,2,5,15,51,188,731,2950,12235,51822,223191,974427,4302645,19181100,86211885,
%T A007317 390248055,1777495635,8140539950,37463689775,173164232965,803539474345,
%U A007317 3741930523740,17481709707825,81912506777200,384847173838501,1812610804416698
%N A007317 Binomial transform of Catalan numbers.
%C A007317 Partial sums of A002212 (the restricted hexagonal polyominoes with n 
               cells). Number of Schroeder paths (i.e. consisting of steps U=(1,
               1),D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) 
               to (2n-2,0), with no peaks at even level. Example: a(3)=5 because 
               among the six Schroeder paths from (0,0) to (4,0) only UUDD has a 
               peak at an even level. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 06 2003
%C A007317 Number of binary trees of weight n where leaves have positive integer 
               weights. Non-commutative Non-associative version of partitions of 
               n. - Michael Somos May 23 2005
%C A007317 Appears also as the number of Euler trees with total weight n (associated 
               with even switching class of matrices of order 2n). - David Garber 
               (garber(AT)hait.ac.il), Sep 19 2005
%C A007317 Number of symmetric hex trees with 2n-1 edges; also number of symmetric 
               hex trees with 2n-2 edges. A hex tree is a rooted tree where each 
               vertex has 0, 1, or 2 children and, when only one child is present, 
               it is either a left child, or a median child, or a right child (name 
               due to an obvious bijection with certain tree-like polyhexes; see 
               the Harary-Read reference). A hex tree is symmetric if it is identical 
               with its reflection in a bisector through the root. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Dec 19 2006
%C A007317 The Hankel transform of [1, 2, 5, 15, 51, 188, ...] is [1, 1, 1, 1, 1, 
               ...], see A000012 ; the Hankel transform of [2, 5, 15, 51, 188, 731, 
               ...] is [2, 5, 13, 34, 89, ...], see A001519 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Dec 19 2006
%C A007317 a(n) = number of 321-avoiding partitions of [n]. A partition is 321-avoiding 
               if the permutation obtained from its canonical form (entries in each 
               block listed in increasing order and blocks listed in increasing 
               order of their first entries) is 321-avoiding. For example, the only 
               partition of [5] that fails to be 321-avoiding is 15/24/3 because 
               the entries 5,4,3 in the permutation 15243 form a 321 pattern. - 
               David Callan (callan(AT)stat.wisc.edu), Jul 22 2008
%C A007317 The sequence 1,1,2,5,15,51,188,... has Hankel transform A001519. [From 
               Paul Barry (pbarry(AT)wit.ie), Jan 13 2009]
%C A007317 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009: 
               (Start)
%C A007317 Equals INVERT transform of A033321: (1, 1, 2, 6, 21, 79, 311,...).
%C A007317 and INVERTi transform of A002212: (1, 3, 10, 36, 137,...). Convolved 
               with A026378, (1, 4, 17, 75, 339,...) = A026376: (1, 6, 30, 144,...)
%C A007317 (End)
%D A007317 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007317 N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and 
               related issues, Discr. Math., 308 (2008), 1209-1221.
%D A007317 R. Bacher and D. Garber, Spindle-configurations of skew lines, submitted
%D A007317 J. Brunvoll et al., Studies of some chemically relevant polygonal systems: 
               mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq 
               15.
%D A007317 S. J. Cyvin et al., Enumeration and classification of benzenoid systems. 
               32. Normal perifusenes with two internal vertices, J. Chem. Inform. 
               Comput. Sci., 32 (1992), 532-540.
%D A007317 S. J. Cyvin et al., Graph-theoretical studies on fluoranthenoids and 
               fluorenoids..., J. Molec. Struct. (Theochem), 285 (1993), 179-185.
%D A007317 S. J. Cyvin et al., Enumeration and classification of certain polygonal 
               systems...: annelated catafusenes, J. Chem. Inform. Comput. Sci., 
               34 (1994), 1174-1180.
%D A007317 I. Pak, Partition identities and geometric bijections. Proc. Amer. Math. 
               Soc. 132 (2004), 3457-3462.
%D A007317 F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. 
               Edinburgh Math. Soc. (2) 17 (1970), 1-13.
%D A007317 Toufik Mansour and Simone Severini, Enumeration of (k,2)-noncrossing 
               partitions, Discrete Math., 308 (2008), 4570-4577.
%H A007317 T. D. Noe, <a href="b007317.txt">Table of n, a(n) for n=1..200</a>
%H A007317 U. Grude, <a href="http://www.tfh-berlin.de/~grude/PapRekursion.pdf">
               Java ist eine Sprache: Rekursive Unterprogramme</a>. See page 4.
%H A007317 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=124">
               Encyclopedia of Combinatorial Structures 124</a>
%H A007317 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A007317 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A007317 David Callan, <a href="http://front.math.ucdavis.edu/0802.2275">Pattern 
               avoidance in "flattened" partitions </a>.
%F A007317 (n+2)*a(n+2) = (6n+4)*a(n+1) - 5n*a(n).
%F A007317 G.f. for sequence doubled: (1/(2*x))*(1+x-(1-x)^(-1)*(1-x^2)^(1/2)*(1-5*x^2)^(1/
               2)).
%F A007317 a(n)= hypergeom([1/2, -n], [2], -4), n=0, 1, 2...; Integral representation 
               as n-th moment of a positive function on a finite interval of the 
               positive half-axis: a(n)=int(x^n*sqrt((5-x)/(x-1))/(2*Pi), x=1..5), 
               n=0, 1, 2... This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), 
               Sep 24 2001
%F A007317 a(1)=1, a(n)=1+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 16 2004
%F A007317 a(n)=sum{k=0..n, (-1)^k*3^(n-k)*binomial(n, k)*binomial(k, floor(k/2))} 
               [offset 0]. - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%F A007317 G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=x-(1-x)(y-y^2) . - Michael 
               Somos May 23 2005
%F A007317 G.f. A(x) satisfies 0=f(x, A(x), A(A(x))) where f(x, y, z)=x(z-z^2)+(x-1)y^2 
               . - Michael Somos May 23 2005
%F A007317 G.f.: (-1+x+(1-6*x+5*x^2)^(1/2))/(2*(-x+x^2)).
%F A007317 G.f. =z*c(z/(1-z))/(1-z) = 1/2 - (1/2)sqrt(1-4z/(1-z)), where c(z)=(1-sqrt(1-4z))/
               (2z) is the Catalan function (follows from Michael Somos' first comment). 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2007
%F A007317 G.f.: 1/(1-2x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-.... (continued 
               fraction). [From Paul Barry (pbarry(AT)wit.ie), Apr 19 2009]
%e A007317 a(3)=5 since {3, (1+2), (1+(1+1)), (2+1), ((1+1)+1)} are the five weighted 
               binary trees of weight 3.
%p A007317 G := (1-sqrt(1-4*z/(1-z)))*1/2: Gser := series(G, z = 0, 30): seq(coeff(Gser, 
               z, n), n = 1 .. 26); - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 12 2007
%t A007317 InverseSeries[Series[(y-y^2)/(1+y-y^2), {y, 0, 24}], x] (* then A(x)=y(x); 
               note that InverseSeries[Series[y-y^2, {y, 0, 24}], x] produces A000108(x) 
               *) - Len Smiley Apr 10 2000
%o A007317 (PARI) {a(n)=local(A); if(n<2, n>0, A=vector(n); for(j=1,n, A[j]=1+sum(k=1,
               j-1, A[k]*A[j-k])); A[n])} /* Michael Somos May 23 2005 */
%o A007317 (PARI) {a(n)=if(n<1, 0, polcoeff(serreverse((x-x^2)/(1+x-x^2)+x*O(x^n)),
               n))} /* Michael Somos May 23 2005 */
%Y A007317 Cf. A000108, A055879. Row sums of absolute values of A091699.
%Y A007317 First column of triangle A104259.
%Y A007317 A033321, A026376, A026378 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 17 2009]
%Y A007317 Sequence in context: A007581 A124303 A073525 this_sequence A153197 A108307 
               A117426
%Y A007317 Adjacent sequences: A007314 A007315 A007316 this_sequence A007318 A007319 
               A007320
%K A007317 easy,nonn,nice
%O A007317 1,2
%A A007317 N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)