A050166 Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• easy

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

Examples

			Triangle begins:
  1;
  1,  2;
  1,  4,  5;
  1,  6, 14, 14;
  1,  8, 27, 48, 42;
  ...
		

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
• 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
• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
• 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.

Crossrefs

Mirror image of A039598.

Programs

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

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

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 *)

PARI

{T(n,k) = 2*(n-k+1)*binomial(2*n+1,k)/(2*n-k+2)}; \\ 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

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)

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

Extensions

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