0,3

Number of ordered trees with n+1 edges, having root of odd degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002

Number of paths of length n with steps U=(1, 1), D=(1, -1), and H=(1, 0), running from (0, 0) to (n, 0) (i.e. grand Motzkin paths of length n). For example, a(3)=7 because we have HHH, HUD, HDU, UDH, DUH, UHD, and DHU. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2003

Binomial transform of A000984, with interpolated zeros. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

Number of leaves in all 0-1-2 trees with n edges, n>0. (A 0-1-2 tree is an ordered tree in which every vertex has at most two children.) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 30 2003

a(n)=number of UDU-free paths of n+1 upsteps (U) and n downsteps (D) that start U. For example, a(2)=3 counts UUUDD, UUDDU, UDDUU. - David Callan (callan(AT)stat.wisc.edu), Aug 18 2004

Diagonal sums of triangle A063007. - Paul Barry (pbarry(AT)wit.ie), Aug 31 2004

Number of ordered ballots from n voters that result in an equal number of votes for candidates A and B in a three candidate election. Ties are counted even when candidates A and B lose the election. For example, a(3)=7 because ballots of the form (voter-1 choice, voter-2 choice, voter-3 choice) that result in equal votes for candidates A and B are the following:(A,B,C), (A,C,B), (B,A,C), (B,C,A), (C,A,B), (C,B,A), and (C,C,C). - Dennis Walsh (dwalsh(AT)mtsu.edu), Oct 08 2004

a(n) = number of weakly increasing sequences (a_1,a_2,...,a_n) with each a_i in [n]={1,2,...,n} and no element of [n] occurring more than twice. For n=3, the sequences are 112, 113, 122, 123, 133, 223, 233. - David Callan (callan(AT)stat.wisc.edu), Oct 24 2004

Note that n divides a(n+1)-a(n). In fact, (a(n+1)-a(n))/n = A007971(n+1). - T. D. Noe (noe(AT)sspectra.com), Mar 16 2005

Row sums of triangle A105868. - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005

a(n) = A111808(n,n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2005

Number of paths of length n with steps U=(1,1), D=(1,-1), and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e. left factors of Motzkin paths) and having no H steps on the x-axis. Example: a(3)=7 because we have UDU, UHD, UHH, UHU, UUD, UUH, and UUU. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 07 2007

G. E. Andrews, "Euler's `exemplum memorabile inductionis fallacis' and $q$-trinomial coefficients", J. Amer. Math. Soc. 3 (1990) 653-669.

E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 78 and 163, #19.

L. Euler, Exemplum Memorabile Inductionis Fallacis, Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 59.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 575.

R. K. Guy, The Second Strong Law of Small Numbers [ Math. Mag, 63(1990) 3-20, esp. 18-19 ]

P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 42.)

V. E. Hoggatt, Jr., and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

E. Pergola, R. Pinzani, S. Rinaldi, and R. A. Sulanke, A bijective approach to the area of generalized Motzkin paths, Adv. Appl. Math., 28, 2002, 580-591.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 74.

L. W. Shapiro et al., The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 6.3.8.

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

G. E. Andrews, Three aspects of partitions

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.

Ed. Pegg, Jr., Number of combinations of n coins when have 3 kinds of coin

Dan Romik, Some formulas for the central trinomial and Motzkin numbers, J. Integer Seqs., Vol. 6, 2003.

T. Sillke, Middle Trinomial Coefficient

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Dennis P. Walsh, The Probablity of a Tie in a Three Candidate Election.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for "core" sequences

Index entries for sequences related to making change.

G.f.: 1/sqrt(1-2*x-3*x^2).

E.g.f.: exp(x) I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 09 2002.

a(n) = 2*A027914(n) - 3^n - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 28 2002

a(n) is asymptotic to d*3^n/sqrt(n) with d around 0.5.. - Benoit Cloitre, Nov 02, 2002

a(n)=((2*n-1)*a(n-1)+3*(n-1)*a(n-2))/n; a(0)=a(1)=1; see paper by Barcucci, Pinzani, and Sprugnoli.

Inverse binomial transform of A000984. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 28 2003

a(n)=sum{k=0..n, C(n, k)C(k, k/2)(1+(-1)^k)/2}; a(n)=sum{k=0..n, (-1)^(n-k)C(n, k)C(2k, k) }. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

a(n) = Sum{k>=0, C(n, 2*k)*C(2*k, k)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 31 2003

a(n)=sum(i+j=n, 0<=j<=i<=n, binomial(n, i)*binomial(i, j)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 06 2004

a(n) = 3* a(n-1) - 2*A005043(n) - Joost Vermeij (joost_vermeij(AT)hotmail.com), Feb 10 2005

a(n) is asymptotic to d*3^n/sqrt(n) with d=sqrt(3/Pi)/2=.488602512... - Alec Mihailovs (alec(AT)mihailovs.com), Feb 24 2005

a(n)=sum{k=0..n, C(n, k)C(k, n-k)}; - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005

a(n) = (-1/4)^n*Sum_{k, 0<=k<=n} = binomial(2k, k)*binomial(2n-2k, n-k)*(-3)^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2005

a(n)=sum{k=0..n, ((1+(-1)^k)/2)*sum{i=0..floor((n-k)/2), C(n, i)C(n-i, i+k)((k+1)/(i+k+1))}}; - Paul Barry (pbarry(AT)wit.ie), Sep 23 2005

a(n)=3^n*sum{j=0..n,(-1/3)^j*C(n,j)*C(2j,j)}; follows from (a) in A027907. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

a(n)=(1/2)^n*sum{j=0..n,3^j*C(n,j)*C(2n-2j,n)}=(3/2)^n*sum{j=0..n,(1/3)^j*C(n,j)*C(2j,n)}; follows from (c) in A027907. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

a(n)=(1/pi)*int(x^n/sqrt((3-x)(1+x)),x,-1,3) is moment representation; - Paul Barry (pbarry(AT)wit.ie), Sep 10 2007

seq(sum('binomial(i, k)*binomial(i-k, k)', 'k'=0..floor(i/2)), i=0..30); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

Table[ CoefficientList[ Series[(1 + x + x^2)^n, {x, 0, n}], x][[ -1]], {n, 0, 27}] (from Robert G. Wilson v)

(PARI) a(n)=if(n<0, 0, polcoeff((1+x+x^2)^n, n))

INVERT transform of A002426 is A007971. Main column of A027907.

Cf. A082758.

Cf. A113302, A113303, A113304, A113305 (divisibility of central trinomial coefficients).

Sequence in context: A018031 A052948 A026325 this_sequence A011769 A087432 A135052

Adjacent sequences: A002423 A002424 A002425 this_sequence A002427 A002428 A002429

easy,nonn,nice,core

njas, Simon Plouffe (plouffe(AT)math.uqam.ca)