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

%I A367505 #43 Jan 23 2024 16:30:08
%S A367505 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 A367505 1,1,43,351,925,925,351,43,1,1,57,637,2401,3675,2401,637,57,1,1,73,
%U A367505 1072,5488,11956,11956,5488,1072,73,1,1,91,1701,11376,33516,47628,33516,11376,1701,91,1
%N A367505 Triangle read by rows: row n gives the h-vector of the n-th halohedron.
%C A367505 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 A367505 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 A367505 Forcey's Hedra Zoo, <a href="https://sforcey.github.io/sf34/hedra.htm#Halohedron">Halohedron</a>.
%F A367505 G.f.: (1 + (1+x)*t)/(2*sqrt(1 - 2*(x+1)*t + (x-1)^2*t^2)) + 1/2.
%F A367505 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 A367505 As a table:
%e A367505   (1),
%e A367505   (1,  1),
%e A367505   (1,  3,  1),
%e A367505   (1,  7,  7,  1),
%e A367505   (1, 13, 27, 13,  1),
%e A367505   (1, 21, 76, 76, 21,  1),
%e A367505   ...
%t A367505 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 A367505 (Sage)
%o A367505 x = polygen(QQ, 'x')
%o A367505 t = x.parent()[['t']].0
%o A367505 F = (1 + (1+x) * t) / (2 * sqrt(1 - 2 * (x+1) * t + (x-1)**2 * t**2)) + 1/2
%o A367505 for poly in F.list(): print(poly.list())
%Y A367505 Row sums are A051960(n-1) for n>=1.
%Y A367505 Alternating sums form an aerated version of A110556.
%Y A367505 Columns k=0-2 give A000012, A002061, A039623(n-1) for n>=2.
%K A367505 nonn,tabl
%O A367505 0,5
%A A367505 _F. Chapoton_, Nov 21 2023