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A002212
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Number of restricted hexagonal polyominoes with n cells.
(Formerly M2850 N1145)
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80
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1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, 751236, 3328218, 14878455, 67030785, 304036170, 1387247580, 6363044315, 29323149825, 135700543190, 630375241380, 2938391049395, 13739779184085, 64430797069375
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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
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
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
Hankel transform of [1,3,10,36,137,543,...]is A000012 =[1,1,1,1,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009: (Start)
Convolved with A026375, (1, 3, 11, 45, 195,...) = A026378: (1, 4, 17, 75,...)
(1, 3, 10, 36, 137,...) convolved with A026375 = A026376: (1, 6, 30, 144,...).
Starting (1, 3, 10, 36,...) = INVERT transform of A007317: (1, 2, 5, 15, 51,...). (End)
Binomial transform of A032357. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]
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REFERENCES
| N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
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.
B. N. Cyvin et al., A class of polygonal systems representing polycyclic conjugated hydrocarbons ..., Monat. f. Chemie, 125 (1994), 1327-1337 (see U(x)).
S. J. Cyvin et al., Number of perifusenes with one internal vertex, Rev. Roumaine Chem., 38 (1993), 65-77.
S. J. Cyvin et al., Graph-theoretical studies on fluoranthenoids and fluorenoids..., J. Molec. Struct. (Theochem), 285 (1993), 179-185.
S. J. Cyvin et al., Enumeration and classification of certain polygonal systems... : annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc. (2) 17 (1970), 1-13.
Lily L. Liu, Positivity of three-term recurrence sequences, Electronic J. Combinatorics, 17 (2010), #R57.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
A. Sapounakis and P. Tsikouras, On k-colored Motzkin words, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.5.
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
Index entries for reversions of series
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FORMULA
| 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
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
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
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
a(n) is asymptotic to c*5^n/n^(3/2) with c=0.63..... - Benoit Cloitre, Jun 23, 2003
Reversion of Sum_{n>0} a(n)x^n = -Sum_{n>0} A001906(n)(-x)^n.
G.f. A(x) satisfies xA(x)^2+(1-x)(1-A(x))=0.
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
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
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
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
a(n+1) = Sum_{k, 0<=k<=n}A097610(n,k)*3^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 02 2007
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]
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]
G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x) (continued fraction); more generally g.f. C(x/(1-x)) where C(x) is the g.f. for the Catalan numbers (A000108). [Joerg Arndt, Mar 18 2011].
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]
For n>=1 a(n)=(1/(2*Pi))*int(x^(n-1)*sqrt((x-1)*(5-x)),x=1..5) - Groux Roland, Mar 16 2011
a(n+1) = [x^n](1-x^2)(1+3*x+x^2)^n [Emanuele Munarini, may 18 2011]
Contribution from Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows, (with 3,2,2,2,...) as the main diagonal:
3, 1, 0, 0, 0, 0,...
1, 2, 1, 0, 0, 0,...
1, 1, 2, 1, 0, 0,...
1, 1, 1, 2, 1, 0,...
1, 1, 1, 1, 2, 0,...
...
Alternatively, let M = the previous matrix but changing the 3 to a 2. Then a(n) = sum of top row terms of M^(n-1). (end)
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MAPLE
| t1 := series( (1-3*x-(1-x)^(1/2)*(1-5*x)^(1/2))/(2*x), x, 50); A002212 := n->coeff(t1, x, n);
a := n->sum(3^(2*i+1-n)*binomial(n, i)*binomial(i, n-i-1), i=ceil((n-1)/2)..n-1)/n;
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:
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
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MATHEMATICA
| InverseSeries[Series[(y)/(1+3*y+y^2), {y, 0, 24}], x] (* then A(x)=y(x) *) - Len Smiley Apr 14 2000
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PROG
| (PARI) a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))/2, n+1)
(PARI) a(n)=if(n<1, n==0, polcoeff(serreverse(x/(1+3*x+x^2)+x*O(x^n)), n)) (from Michael Somos)
(Maxima) makelist(sum(binomial(n, k)*binomial(n-k, k)*3^(n-2*k)/(k+1), k, 0, n/2), n, 0, 24); (* for a(n+1) *) [Emanuele Munarini, may 18 2011]
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CROSSREFS
| Cf. A025238.
First differences of A007317.
Row sums of triangle A104259.
Cf. A000108, A001006.
Cf. A007317, A026376, A026375. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]
Sequence in context: A136576 A129156 A171753 * A149041 A202834 A129247
Adjacent sequences: A002209 A002210 A002211 * A002213 A002214 A002215
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. C. Read (rcread(AT)math.uwaterloo.ca)
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