0,3

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]

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

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

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.

E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).

T. D. Noe, Table of n, a(n) for n=0..200

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

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]

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]

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

InverseSeries[Series[(y)/(1+3*y+y^2), {y, 0, 24}], x] (* then A(x)=y(x) *) - Len Smiley Apr 14 2000

(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)

Cf. A025238.

First differences of A007317.

Row sums of triangle A104259.

Cf. A000108, A001006.

A007317, A026376, A026375 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]

Sequence in context: A026854 A136576 A129156 this_sequence A149041 A129247 A162162

Adjacent sequences: A002209 A002210 A002211 this_sequence A002213 A002214 A002215

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com), R. C. Read (rcread(AT)math.uwaterloo.ca)