

A194460


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


4



1, 4, 18, 82, 370, 1648, 7252, 31582, 136338, 584248, 2488156, 10540484, 44450068, 186715072, 781628008, 3262239862, 13579324498, 56391614632, 233686316428, 966556003132, 3990942300508, 16453094542432, 67733512006168
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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 nl(p), where l(p) is the height of the last peak of p, and the last peak of q has height at least nf(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..n1 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*n1,n1)  2^(2*n1), 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 adnilpotent 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(n1) + 2*binomial(2*n3, n3).  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[2n1, n1]  2^(2n1);
Array[a, 23] (* JeanFrançois Alcover, Jul 27 2018, after Michael Somos *)


PROG

(Sage)
def A194460(n):
if n == 1: return 1
cf = CachedFunction(lambda i, j, n: binomial(n1i+n1j, ni1)binomial(n1i+n1j, nij1))
CP = cartesian_product
return sum(sum(cf(i, j, n)*cf(k, m, n) for k, m in CP([[ni..n], [nj..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*n1, n1)  2^(2*n1))}; /* Michael Somos, Jun 28 2018 */
(Magma) [(n+2)*Binomial(2*n1, n1)  2^(2*n1): n in [1..30]]; // G. C. Greubel, Aug 13 2018


CROSSREFS

Sequence in context: A181610 A264927 A257059 * A356289 A100192 A052913
Adjacent sequences: A194457 A194458 A194459 * A194461 A194462 A194463


KEYWORD

nonn


AUTHOR

Volodymyr Mazorchuk, Aug 24 2011


EXTENSIONS

More terms from D. S. McNeil, Aug 25 2011


STATUS

approved



