A194460 - OEIS

%I #61 Sep 08 2022 08:45:58

%S 1,4,18,82,370,1648,7252,31582,136338,584248,2488156,10540484,

%T 44450068,186715072,781628008,3262239862,13579324498,56391614632,

%U 233686316428,966556003132,3990942300508,16453094542432,67733512006168

%N a(n) is the number of basic ideals in the standard Borel subalgebra of the untwisted affine Lie algebra sl_n.

%C 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.

%C From _Per Alexandersson_, May 26 2018: (Start)

%C a(n) is also equal to the number of circular arc digraphs on n vertices.

%C 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).

%C The subset of such sequences such that b(n)=0 is given by the Catalan numbers, A000108. (End)

%C 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

%H G. C. Greubel, <a href="/A194460/b194460.txt">Table of n, a(n) for n = 1..1000</a>

%H Per Alexandersson, Svante Linusson, Samu Potka, <a href="https://arxiv.org/abs/1903.01327">The cyclic sieving phenomenon on circular Dyck paths</a>, arXiv:1903.01327 [math.CO], 2019.

%H Per Alexandersson and Greta Panova, <a href="https://arxiv.org/abs/1705.10353">LLT polynomials, chromatic quasisymmetric functions and graphs with cycles</a>, arXiv:1705.10353 [math.CO], 2017. See Lemma 5.

%H K. Baur and V. Mazorchuk; <a href="http://arxiv.org/abs/1108.3659">Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras</a>, arXiv:1108.3659 [math.RA], 2011.

%F 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

%F 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

%e 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

%t a[n_] := (n+2) Binomial[2n-1, n-1] - 2^(2n-1);

%t Array[a, 23] (* _Jean-François Alcover_, Jul 27 2018, after _Michael Somos_ *)

%o (Sage)

%o def A194460(n):

%o     if n == 1: return 1

%o     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))

%o     CP = cartesian_product

%o     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]]))

%o # _D. S. McNeil_, Aug 25 2011

%o (PARI) {a(n) = if( n<1, 0, (n+2) * binomial(2*n-1, n-1) - 2^(2*n-1))}; /* _Michael Somos_, Jun 28 2018 */

%o (Magma) [(n+2)*Binomial(2*n-1, n-1) - 2^(2*n-1): n in [1..30]]; // _G. C. Greubel_, Aug 13 2018

%K nonn

%O 1,2

%A _Volodymyr Mazorchuk_, Aug 24 2011

%E More terms from _D. S. McNeil_, Aug 25 2011