%I A002426 M2673 N1070
%S A002426 1,1,3,7,19,51,141,393,1107,3139,8953,25653,73789,212941,616227,
%T A002426 1787607,5196627,15134931,44152809,128996853,377379369,1105350729,
%U A002426 3241135527,9513228123,27948336381,82176836301,241813226151
%N A002426 Central trinomial coefficients: largest coefficient of (1+x+x^2)^n.
%C A002426 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
%C A002426 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
%C A002426 Binomial transform of A000984, with interpolated zeros. - Paul Barry 
               (pbarry(AT)wit.ie), Jul 01 2003
%C A002426 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
%C A002426 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
%C A002426 Diagonal sums of triangle A063007. - Paul Barry (pbarry(AT)wit.ie), Aug 
               31 2004
%C A002426 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
%C A002426 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
%C A002426 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
%C A002426 Row sums of triangle A105868. - Paul Barry (pbarry(AT)wit.ie), Apr 23 
               2005
%C A002426 a(n) = A111808(n,n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 17 2005
%C A002426 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
%C A002426 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008: 
               (Start)
%C A002426 Equals right border of triangle A152227 and starting with offset 1 =
%C A002426 row sums of triangle A152227. (End)
%C A002426 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 07 2009: 
               (Start)
%C A002426 Starting with offset 1 = iterates of M * [1,1,1,...] where M = a tridiagonal
%C A002426 matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the 
               super and subdiagonals. (End)
%C A002426 Hankel transform is 2^n. [From Paul Barry (pbarry(AT)wit.ie), Aug 05 
               2009]
%D A002426 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002426 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002426 G. E. Andrews, "Euler's `exemplum memorabile inductionis fallacis' and 
               $q$-trinomial coefficients", J. Amer. Math. Soc. 3 (1990) 653-669.
%D A002426 E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. 
               A, Vol. 2, 1991, No. 3-4, pp. 249-279.
%D A002426 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A002426 F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 
               (1999) 73-112.
%D A002426 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 78 and 163, #19.
%D A002426 L. Euler, Exemplum Memorabile Inductionis Fallacis, Opera Omnia. Teubner, 
               Leipzig, 1911, Series (1), Vol. 15, p. 59.
%D A002426 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 575.
%D A002426 R. K. Guy, The Second Strong Law of Small Numbers [ Math. Mag, 63(1990) 
               3-20, esp. 18-19 ]
%D A002426 P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 
               vols., 1974-1986. (Vol. 1, p. 42.)
%D A002426 V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal 
               triangles, Fib. Quart., 7 (1969), 341-358, 393.
%D A002426 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%D A002426 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.
%D A002426 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 74.
%D A002426 L. W. Shapiro et al., The Riordan group, Discrete Applied Math., 34 (1991), 
               229-239.
%D A002426 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.
%D A002426 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Example 6.3.8.
%H A002426 T. D. Noe, <a href="b002426.txt">Table of n, a(n) for n = 0..200</a>
%H A002426 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">
               Three aspects of partitions</a>
%H A002426 E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/pdf/math.CO/0407326">
               Congruences for Catalan and Motzkin numbers and related sequences</
               a>, J. Num. Theory 117 (2006), 191-215.
%H A002426 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 A002426 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</
               a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A002426 Ed. Pegg, Jr., <a href="http://www.mathpuzzle.com/coin.html">Number of 
               combinations of n coins when have 3 kinds of coin</a>
%H A002426 Dan Romik, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Some formulas for the central trinomial and Motzkin numbers</a>, 
               J. Integer Seqs., Vol. 6, 2003.
%H A002426 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/
               series008">Middle Trinomial Coefficient</a>
%H A002426 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 A002426 Dennis P. Walsh, <a href="http://www.mtsu.edu/~dwalsh/3votetie.gif">The 
               Probablity of a Tie in a Three Candidate Election</a>.
%H A002426 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CentralTrinomialCoefficient.html">Link to a section of The World 
               of Mathematics.</a>
%H A002426 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrinomialCoefficient.html">Link to a section of The World of Mathematics.</
               a>
%H A002426 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A002426 <a href="Sindx_Mag.html#change">Index entries for sequences related to 
               making change.</a>
%F A002426 G.f.: 1/sqrt(1-2*x-3*x^2).
%F A002426 E.g.f.: exp(x) I_0(2x), where I_0 is Bessel function. - Michael Somos, 
               Sep 09 2002.
%F A002426 a(n) = 2*A027914(n) - 3^n - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Sep 28 2002
%F A002426 a(n) is asymptotic to d*3^n/sqrt(n) with d around 0.5.. - Benoit Cloitre, 
               Nov 02, 2002
%F A002426 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.
%F A002426 Inverse binomial transform of A000984. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 28 2003
%F A002426 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
%F A002426 a(n) = Sum{k>=0, C(n, 2*k)*C(2*k, k)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Dec 31 2003
%F A002426 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
%F A002426 a(n) = 3* a(n-1) - 2*A005043(n) - Joost Vermeij (joost_vermeij(AT)hotmail.com), 
               Feb 10 2005
%F A002426 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
%F A002426 a(n)=sum{k=0..n, C(n, k)C(k, n-k)}; - Paul Barry (pbarry(AT)wit.ie), 
               Apr 23 2005
%F A002426 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
%F A002426 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
%F A002426 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
%F A002426 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
%F A002426 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
%F A002426 G.f.: 1/(1-x-2x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). [From 
               Paul Barry (pbarry(AT)wit.ie), Aug 05 2009]
%F A002426 a(n) = sqrt(-1/3)*(-1)^n*hypergeom([1/2, n+1],[1],4/3) [From Mark van 
               Hoeij (hoeij(AT)math.fsu.edu), Nov 12 2009]
%p A002426 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
%t A002426 Table[ CoefficientList[ Series[(1 + x + x^2)^n, {x, 0, n}], x][[ -1]], 
               {n, 0, 27}] (from Robert G. Wilson v)
%o A002426 (PARI) a(n)=if(n<0,0,polcoeff((1+x+x^2)^n,n))
%Y A002426 INVERT transform of A002426 is A007971. Main column of A027907.
%Y A002426 Cf. A082758.
%Y A002426 Cf. A113302, A113303, A113304, A113305 (divisibility of central trinomial 
               coefficients).
%Y A002426 A152227 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]
%Y A002426 Sequence in context: A018031 A052948 A026325 this_sequence A011769 A087432 
               A135052
%Y A002426 Adjacent sequences: A002423 A002424 A002425 this_sequence A002427 A002428 
               A002429
%K A002426 easy,nonn,nice,core,new
%O A002426 0,3
%A A002426 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)