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%I #55 Sep 08 2022 08:44:58
%S 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,
%T 429,572,429,1,14,90,350,910,1638,2002,1430,1,16,119,544,1700,3808,
%U 6188,7072,4862,1,18,152,798,2907,7752,15504,23256,25194,16796
%N Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.
%C Sometimes called Catalan's triangle, although this term is usually reserved for several other triangles!
%C T is a mirror image of the array in A039598.
%C 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_
%C 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
%C 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
%D 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.
%D Y. Jiang, K. Qiu, R. Qiu, and J. Shen, On the spectrum of the middle-cube, Congressus Numerantium, 195 (2009), 195-204.
%D 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.
%H G. C. Greubel, <a href="/A050166/b050166.txt">Rows n = 0..100 of triangle, flattened</a>
%H E. H. M. Brietzke, <a href="https://doi.org/10.1016/j.disc.2007.08.050">An identity of Andrews and a new method for the Riordan array proof of combinatorial identities</a>, Discrete Math., 308 (2008), 4246-4262.
%H E. Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%H R. K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%H L. W. Shapiro, W.-J. Woan and S. Getu, <a href="https://doi.org/10.1137/0604046">Runs, slides and moments</a>, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.
%F From _Henry Bottomley_, Sep 24 2001: (Start)
%F T(n, k) = C(2n+1, k)*2*(n-k+1)/(2n-k+2) = A039598(n, n-k)
%F 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)
%F 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. - _Philippe Deléham_, Nov 03 2011
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 4, 5;
%e 1, 6, 14, 14;
%e 1, 8, 27, 48, 42;
%e ...
%t 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 *)
%o (PARI) {T(n,k) = 2*(n-k+1)*binomial(2*n+1,k)/(2*n-k+2)}; \\ _G. C. Greubel_, Apr 05 2019
%o (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
%o (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
%o (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
%Y Mirror image of A039598.
%K nonn,tabl,easy
%O 0,3
%A _Clark Kimberling_
%E More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2001
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