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Search: id:A026300
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| A026300 |
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Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1). |
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36
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| 1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 9, 1, 5, 14, 25, 30, 21, 1, 6, 20, 44, 69, 76, 51, 1, 7, 27, 70, 133, 189, 196, 127, 1, 8, 35, 104, 230, 392, 518, 512, 323, 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835, 1, 10, 54, 200, 560, 1242, 2235, 3288, 3915, 3610, 2188
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Right-hand columns have g.f. M^k, where M is g.f. of Motzkin numbers.
Consider a semi-infinite chessboard with squares labeled (i,j), i >= 0, j >= 0; number of king-paths of length j from (0,0) to (i,j), 0 <= i <= j, is T(j,i-j). - Harrie Grondijs (hgrondijs(AT)epo.org), May 27 2005. Cf. A114929, A111808, A114972.
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REFERENCES
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M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259.
J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
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LINKS
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J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
H. Bottomley, Illustration of initial terms
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FORMULA
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T(n, k)=Sum(i=0, Floor(k/2), binomial(n, 2i+n-k)[binomial(2i+n-k, i)-binomial(2i+n-k, i-1)]) - Herbert Kociemba (kociemba(AT)t-online.de), May 27 2004
T(n, k) = A027907(n, k) - A027907(n, k-2), k<=n.
Sum_{k, 0<=k<=n}(-1)^k*T(n,k)=A099323(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 19 2007
Sum(T(n,k) mod 2, 0<=k<=n) = A097357(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 28 2007
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -1, 0, 1, 2 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2009]
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EXAMPLE
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1; 1,1; 1,2,2; 1,3,5,4; 1,4,9,12,9; 1,5,14,25,30,21; ...
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CROSSREFS
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Motzkin numbers (A001006) are T(n, n), other columns of T include A002026, A005322, A005323.
Cf. A020474.
Row sums are in A005773.
Reflected version is in A064189. Cf. A059738.
Sequence in context: A054336 A079956 A140717 this_sequence A099514 A139687 A064581
Adjacent sequences: A026297 A026298 A026299 this_sequence A026301 A026302 A026303
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KEYWORD
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nonn,tabl,nice,new
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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