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A367505
Triangle read by rows: row n gives the h-vector of the n-th halohedron.
0
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
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
KEYWORD
nonn,tabl
AUTHOR
F. Chapoton, Nov 21 2023
STATUS
approved