This site is supported by donations to The OEIS Foundation.



Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A194460 a(n) is the number of basic ideals in the standard Borel subalgebra of the untwisted affine Lie algebra sl_n. 3
1, 4, 18, 82, 370, 1648, 7252, 31582, 136338, 584248, 2488156, 10540484, 44450068, 186715072, 781628008, 3262239862, 13579324498, 56391614632, 233686316428, 966556003132, 3990942300508, 16453094542432, 67733512006168 (list; graph; refs; listen; history; text; internal format)



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


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.


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


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


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

Array[a, 23] (* Jean-Fran├žois Alcover, Jul 27 2018, after Michael Somos *)



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 = CartesianProduct

....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


Sequence in context: A181610 A264927 A257059 * A100192 A052913 A279285

Adjacent sequences:  A194457 A194458 A194459 * A194461 A194462 A194463




Volodymyr Mazorchuk, Aug 24 2011


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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 14 01:33 EST 2019. Contains 329978 sequences. (Running on oeis4.)