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

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, 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, 462, 165, 11, 0, 0, 0, 0, 0, 1, 54, 495, 924, 495, 66, 1, 0, 0, 0, 0, 0, 1, 65, 715

Offset: 1

Author

R. J. Mathar, Jul 31 2016

Keywords

• sign
• tabl
• easy

Examples

			The triangle starts in row n=1 with coefficients 0<=k<n as:
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, 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, 462, 165, 11, 0, 0, 0, 0, 0;
1, 54, 495, 924, 495, 66, 1, 0, 0, 0, 0, 0;
1, 65, 715, 1716, 1287, 286, 13, 0, 0, 0, 0, 0, 0;
		

Links

• Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 10.
• Nan Li, Ehrhart h*-vectors of hypersimplices, Discr. Comp. Geom. 48 (2012) 847-878, Eq. (1.1)

Crossrefs

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).

Programs

Maple

subki := proc(n,r,l)
    local i,t;
    add(t^i,i=0..l-1) ;
    %^n ;
    expand(%) ;
    coeff(%,t,r) ;
end proc:
hstard := proc(d,k,n)
    add((-1)^i*binomial(n,i)*subki(n, (k-i)*d-i,k-i) ,i=0..k-1) ;
end proc:
A275514 := proc(n,k)
    hstard(k,2,n)
end proc: