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
The multiplicities of the eigenvalues of the middle cubes are related to this triangle. The middle cube in Q_3 has eigenvalues -2, -1, 1, 2 with multiplicities 1, 2, 2, 1. The middle cube in Q_5 has eigenvalues -3, -2, -1, 1, 2, 3 with multiplicities 1, 4, 5, 5, 4, 1. The middle cube in Q_7 has eigenvalues -4, -3, -2, -1, 1, 2, 3, 4 with multiplicities 1, 6, 14, 14, 14, 14, 6, 1, etc. - Ke Qiu, Apr 05 2019
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.
Y. Jiang, K. Qiu, R. Qiu, and J. Shen, On the spectrum of the middle-cube, Congressus Numerantium, 195 (2009), 195-204.
A. Nkwanta, Lattice paths and RNA secondary structures, in: Nathaniel Dean, African Americans in Mathematics, AMS and DIMACS, 1997, ISBN 978-0-8218-0678-4, pp. 137-147.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
L. W. Shapiro, W.-J. Woan and S. Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.
FORMULA
From Henry Bottomley, Sep 24 2001: (Start)
T(n, k) = C(2n+1, k)*2*(n-k+1)/(2n-k+2) = A039598(n, n-k)
T(n, k) = T(n-1, k) + 2*T(n-1, k-1) + T(n-1, k-2), with T(0, 0) = 1 and T(n, k) = 0 if n < 0 or n < k. (End)
EXAMPLE
Triangle begins:
1;
1, 2;
1, 4, 5;
1, 6, 14, 14;
1, 8, 27, 48, 42;
...
MATHEMATICA
Table[2*Binomial[2n+1, k]*(n-k+1)/(2*n-k+2), {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 05 2019 *)
PROG
(PARI) {T(n, k) = 2*(n-k+1)*binomial(2*n+1, k)/(2*n-k+2)}; \\ G. C. Greubel, Apr 05 2019
(Magma) [[2*(n-k+1)*Binomial(2*n+1, k)/(2*n-k+2): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019
(Sage) [[2*(n-k+1)*binomial(2*n+1, k)/(2*n-k+2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> 2*(n-k+1)* Binomial(2*n+1, k)/(2*n-k+2) ))); # G. C. Greubel, Apr 05 2019
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2001
STATUS
approved