A194460 -id:A194460 - cp's OEIS frontend

A182186 Number b(n) of basic ideals in the Borel subalgebra of the untwisted affine Lie algebra of type B.

24, 128, 648, 3160, 14984, 69536, 317264, 1427912, 6355080, 28021504, 122586224, 532681648, 2301267408, 9891512000, 42327269792, 180410129576, 766250022536, 3244192404032, 13696322822960, 57673821115088, 242287778611184, 1015664308220864, 4249246138360928

Offset: 2

Jonathan Nilsson, Apr 16 2012

The corresponding sequence for the usual type B Lie algebra is given by the central binomial coefficients A000984.

J. Nilsson, Enumeration of basic ideals in type B
J. Nilsson, Enumeration of Basic Ideals in Type B Lie Algebras, J. Int. Seq. 15 (2012) #12.9.5

Cf. A000984, A194460.

B:=n->(3*n+5)*2^(2*n-2)-(2*(3*n-1))*binomial(2*n-2, n-1): seq(B(n), n=2..30);

a(n) = (3*n+5)*2^(2*n-2)-(2*(3*n-1))*binomial(2*n-2, n-1); \\ Michel Marcus, Aug 18 2013

a(n) = (3*n+5)*2^(2*n-2)-(2*(3*n-1))*binomial(2*n-2, n-1).
a(n) - 8*a(n-1) + 16*a(n-2) = (24/(n-1))*binomial(2*n-6,n-2) for n>3.
-(n-1)*(9*n^2-51*n+76)*a(n) +2*(36*n^3-231*n^2+478*n-295)*a(n-1) -8*(2*n-5)*(9*n^2-33*n+34)*a(n-2)=0. - R. J. Mathar, Oct 27 2017

A200613 Number of quasi-abelian ideals in the affine Lie algebra sl_n^{hat}.

Original entry on oeis.org

1, 3, 11, 44, 183, 774, 3294, 14034, 59711, 253430, 1072506, 4525168, 19036726, 79861404, 334155036, 1394789214, 5808981711, 24143440374, 100156051746, 414762312504, 1714844273586, 7079573497524, 29187378344676, 120180109515204, 494264431607718, 2030539136846844

Offset: 1

N. J. A. Sloane, Nov 19 2011

nonn

Christian Krattenthaler has shown that a(n)=((n+4)/4)*binomial(2*n,n)-3*2^(2*n-3). This implies that a(n)=A194460(n) - A000531(n-1). The latter fact was first empirically observed by D. S. McNeil. [Volodymyr Mazorchuk, Sep 14 2012]

Karin Baur and Volodymyr Mazorchuk, Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras, arXiv:1108.3659 [math.RA]

a(n) = ((n+4)/4)*binomial(2*n,n)-3*2^(2*n-3);

a(n) = ((n+4)/4)*binomial(2*n,n)-3*2^(2*n-3). [Volodymyr Mazorchuk, Sep 14 2012]

A332389 Number A(n,w) of circular Dyck paths with n entries, and width at most w.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 18, 10, 5, 1, 32, 47, 28, 13, 6, 1, 64, 123, 82, 38, 16, 7, 1, 128, 322, 244, 117, 48, 19, 8, 1, 256, 843, 730, 370, 152, 58, 22, 9, 1, 512, 2207, 2188, 1186, 496, 187, 68, 25, 10, 1, 1024, 5778, 6562, 3827, 1648, 622, 222, 78, 28, 11

Offset: 1

Per W. Alexandersson, Feb 10 2020

A(n,w) is the number of circular Dyck paths of size n, and width at most w.
This is also the number of circular area lists, a_1, a_2, ..., a_n such that 0 <= a_i <= w-1, and a_{i+1} < a_i + 1, for all 1 <= i <= n, and the index i is taken modulo n.
The values of w are given by the row index.
A(n,w) is given by summing binomial(2*n - 1, n - 1 - (w+2) k) - binomial(2*n - 1, n + j + (w+2)*k) over k=1..w and k over all integers.

			The table begins as
1,    2,    3,    4,    5, ...
1,    4,    7,    10,   13, ...
1,    8,    18,   28,   38, ...
1,    16,   47,   82,   117, ...
1,    32,   123,  244,  370, ...
...
A(5,3)=123 and a few of the corresponding circular area lists are 00000, 10000,...,12210,...,12222, 22222.

Per Alexandersson, Svante Linusson and Samu Potka, The cyclic sieving phenomenon on circular Dyck paths, arXiv:1903.01327 [math.CO], 2019.
Per Alexandersson, Svante Linusson and Samu Potka, The cyclic sieving phenomenon on circular Dyck paths, Electronic Journal of Combinatorics 26, No.4 (2019).

A194460 is the diagonal.

CircularDyckPaths[n_, w_] := With[{d = w + 2},
   Sum[Binomial[2 n - 1, n - 1 - d s] -
     Binomial[2 n - 1, n + j + d s]
    , {j, w},
    {s, -2 (n + 2), 2 (n + 2)}]
   ];
Table[
CircularDyckPaths[n, w]
, {n, 1, 10}, {w, 1, 10}]

A(n,w) = Sum_{k=-2*(n+2)..2*(n+2)} Sum_{j=1..w} binomial(2n-1, n-1-(w+2)*k) - binomial(2*n-1, n + j + (w+2)*k).

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A182186 Number b(n) of basic ideals in the Borel subalgebra of the untwisted affine Lie algebra of type B.

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A200613 Number of quasi-abelian ideals in the affine Lie algebra sl_n^{hat}.

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A332389 Number A(n,w) of circular Dyck paths with n entries, and width at most w.

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