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Triangle read by rows: row n gives the h-vector of the n-th halohedron.
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%I #43 Jan 23 2024 16:30:08

%S 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,

%T 1,1,43,351,925,925,351,43,1,1,57,637,2401,3675,2401,637,57,1,1,73,

%U 1072,5488,11956,11956,5488,1072,73,1,1,91,1701,11376,33516,47628,33516,11376,1701,91,1

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

%C 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.

%H Jordan Grady Almeter, <a href="https://arxiv.org/abs/2211.02113">P-graph associahedra and hypercube graph associahedra</a>, arXiv:2211.02113 [math.CO], 2022; Ph.D. thesis, North Carolina State University, 2022.

%H Forcey's Hedra Zoo, <a href="https://sforcey.github.io/sf34/hedra.htm#Halohedron">Halohedron</a>.

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

%F 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

%e As a table:

%e (1),

%e (1, 1),

%e (1, 3, 1),

%e (1, 7, 7, 1),

%e (1, 13, 27, 13, 1),

%e (1, 21, 76, 76, 21, 1),

%e ...

%t 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 *)

%o (Sage)

%o x = polygen(QQ, 'x')

%o t = x.parent()[['t']].0

%o F = (1 + (1+x) * t) / (2 * sqrt(1 - 2 * (x+1) * t + (x-1)**2 * t**2)) + 1/2

%o for poly in F.list(): print(poly.list())

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

%Y Alternating sums form an aerated version of A110556.

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

%K nonn,tabl

%O 0,5

%A _F. Chapoton_, Nov 21 2023