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A275514 Triangle read by rows: the coefficient [t^k] of the Ehrhart polynomial of the 2-hypersimplex in dimension n.

%I A275514 #15 May 31 2023 09:50:54
%S A275514 1,1,-1,1,0,0,1,2,1,0,1,5,5,0,0,1,9,15,1,0,0,1,14,35,7,0,0,0,1,20,70,
%T A275514 28,1,0,0,0,1,27,126,84,9,0,0,0,0,1,35,210,210,45,1,0,0,0,0,1,44,330,
%U A275514 462,165,11,0,0,0,0,0,1,54,495,924,495,66,1,0,0,0,0,0,1,65,715
%N A275514 Triangle read by rows: the coefficient [t^k] of the Ehrhart polynomial of the 2-hypersimplex in dimension n.
%H A275514 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 10.
%H A275514 Nan Li, <a href="http://dx.doi.org/10.1007/s00454-012-9452-2">Ehrhart h*-vectors of hypersimplices</a>, Discr. Comp. Geom. 48 (2012) 847-878, Eq. (1.1)
%e A275514 The triangle starts in row n=1 with coefficients 0<=k<n as:
%e A275514 1;
%e A275514 1, -1;
%e A275514 1, 0, 0;
%e A275514 1, 2, 1, 0;
%e A275514 1, 5, 5, 0, 0;
%e A275514 1, 9, 15, 1, 0, 0;
%e A275514 1, 14, 35, 7, 0, 0, 0;
%e A275514 1, 20, 70, 28, 1, 0, 0, 0;
%e A275514 1, 27, 126, 84, 9, 0, 0, 0, 0;
%e A275514 1, 35, 210, 210, 45, 1, 0, 0, 0, 0;
%e A275514 1, 44, 330, 462, 165, 11, 0, 0, 0, 0, 0;
%e A275514 1, 54, 495, 924, 495, 66, 1, 0, 0, 0, 0, 0;
%e A275514 1, 65, 715, 1716, 1287, 286, 13, 0, 0, 0, 0, 0, 0;
%p A275514 subki := proc(n,r,l)
%p A275514     local i,t;
%p A275514     add(t^i,i=0..l-1) ;
%p A275514     %^n ;
%p A275514     expand(%) ;
%p A275514     coeff(%,t,r) ;
%p A275514 end proc:
%p A275514 hstard := proc(d,k,n)
%p A275514     add((-1)^i*binomial(n,i)*subki(n, (k-i)*d-i,k-i) ,i=0..k-1) ;
%p A275514 end proc:
%p A275514 A275514 := proc(n,k)
%p A275514     hstard(k,2,n)
%p A275514 end proc:
%Y A275514 Cf. A010054 (coefficients for the 1-hypersimplex), A258993, A000295 (row sums), A000096 (column k=1), A000332 (k=2), A000579 (k=3), A000581 (k=4), A001287 (k=5).
%K A275514 sign,tabl,easy
%O A275514 1,8
%A A275514 _R. J. Mathar_, Jul 31 2016