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A002426 Central trinomial coefficients: largest coefficient of (1+x+x^2)^n.
(Formerly M2673 N1070)
188
1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, 616227, 1787607, 5196627, 15134931, 44152809, 128996853, 377379369, 1105350729, 3241135527, 9513228123, 27948336381, 82176836301, 241813226151 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of ordered trees with n+1 edges, having root of odd degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch, 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, May 31 2003

Number of lattice paths from (0,0) to (n,n) using steps (2,0), (0,2), (1,1). It appears that 1/sqrt((1-x)^2-4*x^s) is the g.f. for lattice paths from (0,0) to (n,n) using steps (s,0), (0,s), (1,1). [Joerg Arndt, Jul 01 2011]

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2). [Joerg Arndt, Jul 05 2011]

Binomial transform of A000984, with interpolated zeros. - Paul Barry, 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, 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, Aug 18 2004

Diagonal sums of triangle A063007. - Paul Barry, 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 P. Walsh, 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, 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, Mar 16 2005

Row sums of triangle A105868. - Paul Barry, Apr 23 2005

a(n) = A111808(n,n). - Reinhard Zumkeller, 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, Oct 07 2007

Equals right border of triangle A152227; starting with offset 1, the row sums of triangle A152227. [Gary W. Adamson, Nov 29 2008]

Starting with offset 1 = iterates of M * [1,1,1,...] where M = a tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. [Gary W. Adamson, Jan 07 2009]

Hankel transform is 2^n. [From Paul Barry, Aug 05 2009]

a(n) is prime for n=2, 3, and 4, with no others for n<=10^5 (E. W. Weisstein, Mar. 14, 2005). It has apparently not been proved that no [other] prime central trinomials exist. [From Jonathan Vos Post, Mar 19 2010]

a(n) is not divisible by 3 for n whose base 3 representation contains no 2, A005836

a(n) = number of (n-1)-lettered words in the alphabet {1, 2, 3} with as many occurrences of the substring (consecutive subword) [1, 2] as those of [2, 1]. See the papers by Ekhad-Zeilberger and Zeilberger. - N. J. A. Sloane, Jul 05 2012

a(n) = coefficient of x^n in (1+x+x^2)^n. - L. Edson Jeffery, Mar 23 2013

a(n) is the number of ordered pairs (A,B) of subsets of {1,2,...,n} such that i.) A and B are disjoint and ii.) A and B contain the same number of elements.  For example, a(2) = 3 because we have: ({},{}) ; ({1},{2}) ; ({2},{1}). - Geoffrey Critzer, Sep 04 2013

Also central terms of A082601. - Reinhard Zumkeller, Apr 13 2014

REFERENCES

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.

S. Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, 16 (2013), #13.1.3. - From N. J. A. Sloane, Feb 03 2013

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.

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.

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).

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.

Z.-W. Sun, Conjectures involving arithmetical sequences, Number Theory: Arithmetic in Shangrila (eds., S. Kanemitsu, H.-Z. Li and J.-Y. Liu), Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258; http://math.nju.edu.cn/~zwsun/142p.pdf. - From N. J. A. Sloane, Dec 28 2012

LINKS

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

G. E. Andrews, Three aspects of partitions

J. Cigler, Some nice Hankel determinants. Arxiv preprint arXiv:1109.1449, 2011.

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

Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, Arxiv preprint arXiv:1112.6207, 2011.

Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, Arxiv preprint arXiv:1203.6792, 2012. - From N. J. A. Sloane, Oct 03 2012

Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, Arxiv preprint arXiv:1110.6638, 2011

Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, Arxiv preprint arXiv:1203.1476, 2012. - From N. J. A. Sloane, Sep 14 2012

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

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

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.

E. Rowland, R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635, 2013

J. Salas and A. D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279-373, arXiv:0711.1738. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014

T. Sillke, Middle Trinomial Coefficient

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

Zhi-Wei Sun, Conjectures involving combinatorial sequences, Arxiv preprint arXiv:1208.2683, 2012. - From N. J. A. Sloane, Dec 25 2012

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

Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595, 2013

Eric Weisstein's World of Mathematics, Central Trinomial Coefficient

Eric Weisstein's World of Mathematics, Trinomial Coefficient

D. Zeilberger, Analogs of the Richard Stanley Amer. Math. Monthly Problem 11610 for ALL pairs of words of length, 2, in an alphabet of, 3 letters. See Proposition 5

Index entries for "core" sequences

Index entries for sequences related to making change.

FORMULA

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, 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, 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, Jul 01 2003

a(n) = Sum{k>=0, C(n, 2*k)*C(2*k, k)}. - Philippe Deléham, Dec 31 2003

a(n)=sum(i+j=n, 0<=j<=i<=n, binomial(n, i)*binomial(i, j)) - Benoit Cloitre, 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, Apr 23 2005

a(n) = (-1/4)^n*Sum_{k, 0<=k<=n} = binomial(2k, k)*binomial(2n-2k, n-k)*(-3)^k . - Philippe Deléham, 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, 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, Sep 10 2007

G.f.: 1/(1-x-2x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). [From Paul Barry, Aug 05 2009]

a(n) = sqrt(-1/3)*(-1)^n*hypergeom([1/2, n+1],[1],4/3) [From Mark van Hoeij, Nov 12 2009]

a(n) = (1/pi)*int((1+2*x)^n/sqrt(1-x^2),x,-1,1) = (1/pi)*int((1+2*cos(t))^n,t,0,pi). - Eli Wolfhagen, Feb 01 2011

In general, G.f.: 1/sqrt(1-2*a*x+(x^2)*((a^2)-4*b)) = 1/(1-a*x)*(1 - 2*(x^2)*b/(G(0)*(a*x-1) + 2*(x^2)*b)); G(k)= 1 - a*x - (x^2)*b/G(k+1); for G.f.: 1/sqrt(1-2*x-3*(x^2))=1/(1-x)*(1 - 2*(x^2)/(G(0)*(x-1) + 2*(x^2))); G(k)= 1 - x - (x^2)/G(k+1), a=1,b=1; (continued fraction). - Sergei N. Gladkovskii, Dec 08 2011

a(n) = sum{k=0..floor(n/3), (-1)^k*binomial(2n-3k-1, n-3k)*binomial(n, k)}. - Gopinath A. R., Feb 10 2012

G.f.: A(x) = x*B’(x)/B(x) where B(x) satisfies B(x) = x*(1+B(x)+B(x)^2). - Vladimir Kruchinin, Feb 03 2013

G.f.:  G(0), where G(k)= 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013

E.g.f.: Sum_{k>=0} exp(x)*x^k/k!*x^k/k!. - Geoffrey Critzer, Sep 04 2013

G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) / (1-x)^(2*n+1). - Paul D. Hanna, Sep 21 2013

MAPLE

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

MATHEMATICA

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

a=b=1; Join[{a, b}, Table[c=((2n-1)b + 3(n-1)a)/n; a=b; b=c; c, {n, 2, 100}]]; Table[Sqrt[-3]^n LegendreP[n, 1/Sqrt[-3]], {n, 0, 26}] (* Wouter Meeussen, Feb 16 2013 *)

PROG

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

(Maxima) trinomial(n, k):=coeff(expand((1+x+x^2)^n), x, k);

makelist(trinomial(n, n), n, 0, 12); [Emanuele Munarini, Mar 15 2011]

(PARI) /* as lattice paths: same as in A092566 but use */

steps=[[2, 0], [0, 2], [1, 1]];

/* Joerg Arndt, Jul 01 2011 */

(MAGMA) P<x>:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2)^n)): n in [0..26] ]; // Bruno Berselli, Jul 05 2011

(Haskell)

a002426 n = a027907 n n  -- Reinhard Zumkeller, Jan 22 2013

(PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m) / (1-x+x*O(x^n))^(2*m+1)), n)} \\ Paul D. Hanna, Sep 21 2013

CROSSREFS

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

Cf. A082758, A152227, A102445.

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

Cf. A097893 (partial sums).

Sequence in context: A018031 A052948 A026325 * A011769 A087432 A135052

Adjacent sequences:  A002423 A002424 A002425 * A002427 A002428 A002429

KEYWORD

easy,nonn,nice,core,changed

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified April 24 01:26 EDT 2014. Contains 240947 sequences.