A367505 Triangle read by rows: row n gives the h-vector of the n-th halohedron.

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 27, 13, 1, 1, 21, 76, 76, 21, 1, 1, 31, 175, 300, 175, 31, 1, 1, 43, 351, 925, 925, 351, 43, 1, 1, 57, 637, 2401, 3675, 2401, 637, 57, 1, 1, 73, 1072, 5488, 11956, 11956, 5488, 1072, 73, 1, 1, 91, 1701, 11376, 33516, 47628, 33516, 11376, 1701, 91, 1

Offset

0,5

Comments

Theorem 6.1.11 in Almeter's thesis gives the f-vector generating series. Then replacing x with x-1 gives the h-vector generating series.

Links

Table of n, a(n) for n=0..65.

Jordan Grady Almeter, P-graph associahedra and hypercube graph associahedra, arXiv:2211.02113 [math.CO], 2022; Ph.D. thesis, North Carolina State University, 2022.

Forcey's Hedra Zoo, Halohedron.

Formula

G.f.: (1 + (1+x)*t)/(2*sqrt(1 - 2*(x+1)*t + (x-1)^2*t^2)) + 1/2.

T(0,0) = 1; T(n,k) = binomial(n-1,n-k)*binomial(n,n-k)+binomial(n-1,n-k-1)^2. - Detlef Meya, Nov 23 2023

Example

As a table:
  (1),
  (1,  1),
  (1,  3,  1),
  (1,  7,  7,  1),
  (1, 13, 27, 13,  1),
  (1, 21, 76, 76, 21,  1),
  ...

Mathematica

T[0, 0]:=1; T[n_, k_]:= Binomial[n-1, n-k]*Binomial[n, n-k]+Binomial[n-1, n-k-1]^2; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Nov 23 2023 *)

Prog

(Sage)
x = polygen(QQ, 'x')
t = x.parent()[['t']].0
F = (1 + (1+x) * t) / (2 * sqrt(1 - 2 * (x+1) * t + (x-1)**2 * t**2)) + 1/2
for poly in F.list(): print(poly.list())

Crossrefs

Row sums are A051960(n-1) for n>=1.

Alternating sums form an aerated version of A110556.

Columns k=0-2 give A000012, A002061, A039623(n-1) for n>=2.

Sequence in context: A132733 A347971 A082039 * A176331 A157836 A205497

Adjacent sequences: A367502 A367503 A367504 * A367506 A367507 A367508

Keyword

nonn,tabl

Author

F. Chapoton, Nov 21 2023

Status

approved