%I A002212 M2850 N1145
%S A002212 1,1,3,10,36,137,543,2219,9285,39587,171369,751236,3328218,
%T A002212 14878455,67030785,304036170,1387247580,6363044315,29323149825,
%U A002212 135700543190,630375241380,2938391049395,13739779184085,64430797069375
%N A002212 Number of restricted hexagonal polyominoes with n cells.
%C A002212 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,0) with 
               no peaks at odd level. Example: a(2)=3 because we have UUDD, UHD 
               and HH. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003
%C A002212 Number of 3-Motzkin paths of length n-1 (i.e. lattice paths from (0,0) 
               to (n-1,0) that do not go below the line y=0 and consist of steps 
               U=(1,1), D=(1,-1) and three types of steps H=(1,0)). Example: a(4)=36 
               because we have 27 HHH paths, 3 HUD paths, 3 UHD paths and 3 UDH 
               paths. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 22 2004
%C A002212 Number of rooted, planar trees having edges weighted by strictly positive 
               natural integers (multi-trees) with weight-sum n. - Roland Bacher 
               (Roland.Bacher(AT)ujf-grenoble.fr), Feb 28 2005
%C A002212 Number of skew Dyck paths of semilength n. A skew Dyck path is a path 
               in the first quadrant which begins at the origin, ends on the x-axis, 
               consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) 
               so that up and left steps do not overlap. The length of the path 
               is defined to be the number of its steps. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 10 2007
%C A002212 Hankel transform of [1,3,10,36,137,543,...]is A000012 =[1,1,1,1,...] 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
%C A002212 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009: 
               (Start)
%C A002212 Convolved with A026375, (1, 3, 11, 45, 195,...) = A026378: (1, 4, 17, 
               75,...)
%C A002212 (1, 3, 10, 36, 137,...) convolved with A026375 = A026376: (1, 6, 30, 
               144,...).
%C A002212 Starting (1, 3, 10, 36,...) = INVERT transform of A007317: (1, 2, 5, 
               15, 51,...). (End)
%C A002212 Binomial transform of A032357. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 17 2009]
%D A002212 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002212 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002212 N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and 
               related issues, Discr. Math., 308 (2008), 1209-1221.
%D A002212 J. Brunvoll et al., Studies of some chemically relevant polygonal systems: 
               mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq 
               14.
%D A002212 B. N. Cyvin et al., A class of polygonal systems representing polycyclic 
               conjugated hydrocarbons ..., Monat. f. Chemie, 125 (1994), 1327-1337 
               (see U(x)).
%D A002212 S. J. Cyvin et al., Number of perifusenes with one internal vertex, Rev. 
               Roumaine Chem., 38 (1993), 65-77.
%D A002212 S. J. Cyvin et al., Graph-theoretical studies on fluoranthenoids and 
               fluorenoids..., J. Molec. Struct. (Theochem), 285 (1993), 179-185.
%D A002212 S. J. Cyvin et al., Enumeration and classification of certain polygonal 
               systems... : annelated catafusenes, J. Chem. Inform. Comput. Sci., 
               34 (1994), 1174-1180.
%D A002212 F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. 
               Edinb. Math. Soc. (2) 17 (1970), 1-13.
%D A002212 E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).
%H A002212 T. D. Noe, <a href="b002212.txt">Table of n, a(n) for n=0..200</a>
%H A002212 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 A002212 A. Sapounakis and P. Tsikouras, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/">On k-colored Motzkin words</a>, Journal of Integer Sequences, 
               Vol. 7 (2004), Article 04.2.5.
%H A002212 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 
               3 (2000), #00.1.
%H A002212 <a href="Sindx_Res.html#revert">Index entries for reversions of series</
               a>
%F A002212 Also: a(0)=1, for n>0: a(n)=Sum(Sum a(i)a(j-i), (i=0, .., j)), (j=0, 
               .., n-1). G.f.: A(x)=1+xA(x)^2/(1-x). - Mario Catalani (mario.catalani(AT)unito.it), 
               Jun 19 2003
%F A002212 a(n)=sum(3^(2i+1-n)*binomial(n, i)*binomial(i, n-i-1), i=ceil((n-1)/2)..n-1)/
               n; - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 23 2002
%F A002212 a(n)=sum(binomial(2k, k)*binomial(n-1, k-1)/(k+1), k=1..n), i.e. binomial 
               transform of the Catalan numbers 1, 2, 5, 14, 42, ... (A000108). 
               a(n)=sum(3^(n-1-2*k)*binomial(2k, k)*binomial(n-1, 2k)/(k+1), k=0..floor((n-1)/
               2)); - EmericDeutsch(AT)msn.com (deutsch(AT)duke.poly.edu), Aug 05 
               2002
%F A002212 a(1)=1, a(n)=(3(2n-1)*a(n-1)-5(n-2)*a(n-2))/(n+1) for n > 1 - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Dec 18 2002
%F A002212 a(n) is asymptotic to c*5^n/n^(3/2) with c=0.63..... - Benoit Cloitre, 
               Jun 23, 2003
%F A002212 Reversion of Sum_{n>0} a(n)x^n = -Sum_{n>0} A001906(n)(-x)^n.
%F A002212 G.f. A(x) satisfies xA(x)^2+(1-x)(1-A(x))=0.
%F A002212 G.f.: (1-x-sqrt(1-6x+5x^2))/(2x). a(n)=3a(n-1)+Sum[a(k)a(n-k-1)], k=1, 
               ..., n-2, for n>1 - John W. Layman (layman(AT)math.vt.edu), Feb 22 
               2001
%F A002212 The Hankel transform of this sequence gives A001519 = 1, 2, 5, 13, 34, 
               89, ... E.g. Det([1, 1, 3, 10, 36; 1, 3, 10, 36, 137; 3, 10, 36, 
               137, 543; 10, 36, 137, 543, 2219; 36, 137, 543, 2219, 9285 ])= 34. 
               - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 25 2004
%F A002212 a(m+n+1) = Sum_{k, k>=0} A091965(m, k)*A091965(n, k) = A091965(m+n, 0) 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005
%F A002212 a(n+1)=Sum(2^(n-k)*M(k)*binom(n,k), k=0..n), where M(k)=A001006(k) are 
               the Motzkin numbers (from here it follows that a(n+1) and M(n) have 
               the same parity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 
               10 2007
%F A002212 a(n+1) = Sum_{k, 0<=k<=n}A097610(n,k)*3^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 02 2007
%F A002212 G.f.: 1/(1-x/(1-x-x/(1-x/(1-x-x/(1-x/(1-x-x/(1-... (continued fraction). 
               [From Paul Barry (pbarry(AT)wit.ie), May 16 2009]
%F A002212 G.f.: (1-x)/(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), Oct 17 
               2009]
%F A002212 a(n) = -5^(1/2)/(10*(n+1)) * (5*hypergeom([1/2, n],[1],4/5)-3*hypergeom([1/
               2, n+1],[1],4/5)) (for n>0) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), 
               Nov 12 2009]
%p A002212 t1 := series( (1-3*x-(1-x)^(1/2)*(1-5*x)^(1/2))/(2*x), x, 50); A002212 
               := n->coeff(t1,x,n);
%p A002212 a := n->sum(3^(2*i+1-n)*binomial(n,i)*binomial(i,n-i-1),i=ceil((n-1)/
               2)..n-1)/n;
%p A002212 a[1] := 1: for n from 2 to 50 do a[n] := (3*(2*n-1)*a[n-1]-5*(n-2)*a[n-2])/
               (n+1) od:
%p A002212 Z:=(1-4*z*sqrt(1-6*z+5*z^2))/8: Zser:=series(Z, z=0, 32): seq(coeff(Zser, 
               z, n), n=3..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 01 2007
%t A002212 InverseSeries[Series[(y)/(1+3*y+y^2), {y, 0, 24}], x] (* then A(x)=y(x) 
               *) - Len Smiley Apr 14 2000
%o A002212 (PARI) a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))/2,n+1)
%o A002212 (PARI) a(n)=if(n<1,n==0,polcoeff(serreverse(x/(1+3*x+x^2)+x*O(x^n)),n)) 
               (from Michael Somos)
%Y A002212 Cf. A025238.
%Y A002212 First differences of A007317.
%Y A002212 Row sums of triangle A104259.
%Y A002212 Cf. A000108, A001006.
%Y A002212 A007317, A026376, A026375 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 17 2009]
%Y A002212 Sequence in context: A026854 A136576 A129156 this_sequence A149041 A129247 
               A162162
%Y A002212 Adjacent sequences: A002209 A002210 A002211 this_sequence A002213 A002214 
               A002215
%K A002212 nonn,easy,nice
%O A002212 0,3
%A A002212 N. J. A. Sloane (njas(AT)research.att.com), R. C. Read (rcread(AT)math.uwaterloo.ca)