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A050166 Triangle T(n,k)=M(2n,k,-1), 0<=k<=n, n >= 0, array M as in A050144. 7
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; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

The diagonals of this triangle are self-convolutions of the main diagonal A000108(n+1) : 1, 2, 5, 14, 42, 132, 429, . . . - Philippe Deléham, May 25 2005

REFERENCES

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.

L. W. Shapiro, W.-J. Woan and S. Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.

LINKS

Table of n, a(n) for n=0..56.

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

FORMULA

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, Sep 24 2001

Sum_{0<=k<=n} T(n,k)*x^k = A000012(n), A001700(n), A194723(n+1), A194724(n+1), A194725(n+1), A194726(n+1), A195727(n+1), A194728(n+1), A194729(n+1), A194730(n+1) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - From Philippe Deléham, Nov 03 2011

EXAMPLE

Rows: {1}; {1,2}; {1,4,5}; ...

CROSSREFS

Mirror image of A039598.

Sequence in context: A237274 A038730 A188106 * A124959 A081281 A108198

Adjacent sequences:  A050163 A050164 A050165 * A050167 A050168 A050169

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2001

STATUS

approved

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Last modified September 26 00:10 EDT 2017. Contains 292500 sequences.