OFFSET
0,2
COMMENTS
By reading paths backward, the UUD in the name could be replaced by DDU.
Or, triangular array T read by rows: T(n,k)=P(2n,n,4k), where P(n,k,c)=number of vectors (x(1),x(2,),...,x(n)) of k 1's and n-k 0's such that x(i)=x(n+1-i) for exactly c values of i. P(n,k,n) counts palindromes.
In nuclear magnetic resonance of n coupled spin-1/2 nuclides, T(n,k) is the number of zero-quantum transitions with combination index k. See the [Sykora (2007)] link, containing also yet another interpretation in terms of pairs of binary n-tuples. - Stanislav Sykora, Apr 27 2012
Let u - (u_1, u_2, u_3, ..., u_{2n}) be a binary vector containing n 0's and n 1's. Define a mismatch to be an adjacent pair (u_{2i-1}, u_{2i}) which is neither 0,1 nor 1,0 (think "socks"). Then T(n,k) = number of u's with k mismatches. - N. J. A. Sloane, Nov 03 2017 following an email from Bill Gosper
From Colin Defant, Sep 16 2018: (Start)
Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n] with k valleys such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n] with k valleys such that s(pi) avoids the patterns 132, 312, and 321.
T(n,k) is the number of permutations of [n] with k valleys that avoid the patterns 1342, 3142, 3412, and 3421. (End)
LINKS
Stanislav Sykora, Table of n, a(n) for n = 0..2600
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.
Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2.
Stanislav Sykora, Triangle T(n,k) for rows n = 0..100
Stanislav Sykora, p-Quantum Transitions and a Combinatorial Identity, Stan's Library II, 2007, Identity (1) for p=0.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
FORMULA
T(n, k) = binomial(n, 2*k)*2^(n-2*k)*binomial(2*k, k).
G.f.: (1-4*x+4*x^2*(1-y))^(-1/2) = Sum_{n>=0, k>=0} a(n, k)*x^n*y^k.
EXAMPLE
Table begins
n | k=0 1 2 3
--+-------------------
0 | 1
1 | 2
2 | 4 2
3 | 8 12
4 | 16 48 6
5 | 32 160 60
6 | 64 480 360 20
7 | 128 1344 1680 280
...
a(2,1)=2 because UUDD, DUUD each have one UUD.
MATHEMATICA
Table[Binomial[n, 2k]2^(n-2k)Binomial[2k, k], {n, 0, 15}, {k, 0, n/2}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
Additional comments from David Callan, Aug 28 2004
STATUS
approved