OFFSET
1,2
COMMENTS
a(n) also equals the number of pairs (p,q) of Dyck paths of semilength n, such that the first peak of q has height at least n-l(p), where l(p) is the height of the last peak of p, and the last peak of q has height at least n-f(p), where f(p) is the height of the first peak of p.
From Per Alexandersson, May 26 2018: (Start)
a(n) is also equal to the number of circular arc digraphs on n vertices.
a(n) is equal to the number of lists b(1),b(2),...,b(n) such that 0 <= b(i) < n and b(i)-1 <= b(i+1) for i=1..n-1 and b(n)-1 <= b(1).
The subset of such sequences such that b(n)=0 is given by the Catalan numbers, A000108. (End)
Christian Krattenthaler has shown that a(n) = (n+2)*binomial(2*n-1,n-1) - 2^(2*n-1), which also implies the above recursion observed by D. S. McNeil. - Volodymyr Mazorchuk, Aug 26 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Per Alexandersson, Svante Linusson, Samu Potka, The cyclic sieving phenomenon on circular Dyck paths, arXiv:1903.01327 [math.CO], 2019.
Per Alexandersson and Greta Panova, LLT polynomials, chromatic quasisymmetric functions and graphs with cycles, arXiv:1705.10353 [math.CO], 2017. See Lemma 5.
K. Baur and V. Mazorchuk; Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras, arXiv:1108.3659 [math.RA], 2011.
FORMULA
It appears that the sequence is given by a(1)=1, a(n) = 4*a(n-1) + 2*binomial(2*n-3, n-3). - D. S. McNeil, Aug 25 2011
0 = a(n)*(+2304*a(n+1) -3744*a(n+2) +1464*a(n+3) -168*a(n+4)) +a(n+1)*(-96*a(n+1) +1192*a(n+2) -730*a(n+3) +102*a(n+4)) +a(n+2)*(-78*a(n+2) +99*a(n+3) -19*a(n+4)) +a(n+3)*(-3*a(n+3) +a(n+4)) for all n>0. - Michael Somos, Jun 28 2018
EXAMPLE
G.f. = x + 4*x^2 + 18*x^3 + 82*x^4 + 370*x^5 + 1648*x^6 + 7252*x^7 + 31582*x^8 + ... - Michael Somos, Jun 28 2018
MATHEMATICA
a[n_] := (n+2) Binomial[2n-1, n-1] - 2^(2n-1);
Array[a, 23] (* Jean-François Alcover, Jul 27 2018, after Michael Somos *)
PROG
(Sage)
def A194460(n):
if n == 1: return 1
cf = CachedFunction(lambda i, j, n: binomial(n-1-i+n-1-j, n-i-1)-binomial(n-1-i+n-1-j, n-i-j-1))
CP = cartesian_product
return sum(sum(cf(i, j, n)*cf(k, m, n) for k, m in CP([[n-i..n], [n-j..n]])) for i, j in CP([[1..n], [1..n]]))
# D. S. McNeil, Aug 25 2011
(PARI) {a(n) = if( n<1, 0, (n+2) * binomial(2*n-1, n-1) - 2^(2*n-1))}; /* Michael Somos, Jun 28 2018 */
(Magma) [(n+2)*Binomial(2*n-1, n-1) - 2^(2*n-1): n in [1..30]]; // G. C. Greubel, Aug 13 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Volodymyr Mazorchuk, Aug 24 2011
EXTENSIONS
More terms from D. S. McNeil, Aug 25 2011
STATUS
approved