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Search: id:A050166
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| A050166 |
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Triangle T(n,k)=M(2n,k,-1), 0<=k<=n, n >= 0, array M as in A050144. |
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+0
6
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| 1, 1, 2, 1, 4, 5, 1, 6, 14, 14, 1, 8, 27, 48, 42, 1, 10, 44, 110, 165, 132, 1, 12, 65, 208, 429, 572, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 16, 119, 544, 1700, 3808, 6188, 7072, 4862, 1, 18, 152, 798, 2907, 7752, 15504, 23256, 15194, 16796, 1, 20
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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Sometimes called Catalan's triangle, although this term is usually reserved for several other triangles!
T is a mirror image of the array in A039598.
Given (1) = row 0, then the sum of terms with alternating signs in row r of A050166 = (-1)^r * A000108(n); where A000108 = 1, 1, 2, 5, 14, 42...the Catalan numbers. - Herb Conn, HCR 83, Box 93, Custer, SD 57730
The diagonals of this triangle are self-convolutions of the main diagonal A000108(n+1) : 1, 2, 5, 14, 42, 132, 429, . . . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 25 2005
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REFERENCES
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B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
E. H. M. Brietzke, An identity of Andrews ..., Discrete Math., 308 (2008), 4246-4262.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
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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R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
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a(n, k) = C(2n+1, k)*2*(n-k+1)/(2n-k+2) = A039598(n, n-k) = a(n-1, k)+2*a(n-1, k-1)+a(n-1, k-2) [with a(0, 0) = 1 and a(n, k) = 0 if n<0 or n<k]. - Henry Bottomley (se16(AT)btinternet.com), Sep 24 2001
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EXAMPLE
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Rows: {1}; {1,2}; {1,4,5}; ...
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CROSSREFS
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Sequence in context: A021983 A161135 A038730 this_sequence A124959 A081281 A108198
Adjacent sequences: A050163 A050164 A050165 this_sequence A050167 A050168 A050169
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2001
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