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A002426
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Central trinomial coefficients: largest coefficient of (1+x+x^2)^n.
(Formerly M2673 N1070)
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165
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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; internal format)
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OFFSET
| 0,3
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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 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, 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]
Contribution from Gary W. Adamson, Jan 07 2009: (Start)
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. (End)
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.
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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.
J. Cigler, Some nice Hankel determinants. Arxiv preprint arXiv:1109.1449, 2011.
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.
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
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.
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.
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LINKS
| 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, Central Trinomial Coefficient
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for "core" sequences
Index entries for sequences related to making change.
Eric W. Weisstein, Central Trinomial Coefficient. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 19 2010]
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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 (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.rs), 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, 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, 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 (hoeij(AT)math.fsu.edu), 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
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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
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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}]]
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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
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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).
Sequence in context: A018031 A052948 A026325 * A011769 A087432 A135052
Adjacent sequences: A002423 A002424 A002425 * A002427 A002428 A002429
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KEYWORD
| easy,nonn,nice,core,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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