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A050166
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Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.
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8
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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, 25194, 16796
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OFFSET
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0,3
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COMMENTS
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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
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
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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.
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.
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LINKS
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FORMULA
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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)
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
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EXAMPLE
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Triangle begins:
1;
1, 2;
1, 4, 5;
1, 6, 14, 14;
1, 8, 27, 48, 42;
...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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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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STATUS
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approved
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