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%I #28 Dec 13 2019 05:59:02
%S 1,1,2,1,3,2,1,4,5,4,1,5,8,9,2,1,6,13,22,15,8,1,7,18,41,40,39,2,1,8,
%T 25,66,103,112,45,16,1,9,32,107,202,275,182,129,6,1,10,41,158,381,730,
%U 685,688,149,32,1,11,50,219,636,1589,2036,2525,844,243,2,1,12,61,304,1033,3000,5153,7488,5221,2090,369,64,1
%N T(n,k) = number of 0..k arrays x(0..n-1) of n elements with zero (n-1)-st difference.
%C Table starts
%C 1 1 1 1 1 1 1 1 1 1 1
%C 2 3 4 5 6 7 8 9 10 11 12
%C 2 5 8 13 18 25 32 41 50 61 72
%C 4 9 22 41 66 107 158 219 304 403 516
%C 2 15 40 103 202 381 636 1033 1550 2287 3212
%C 8 39 112 275 730 1589 3000 5181 8350 13871 21588
%C 2 45 182 685 2036 5153 11370 23035 43284 76523 129052
%C 16 129 688 2525 7488 18809 52166 121921 253768 484977 867086
%C 6 149 844 5221 19262 68813 194818 514113 1171190 2531421 5019770
%C 32 243 2090 13897 62772 256859 841122 2347671 6169890 14503751 31169760
%C T(n,k) is the number of integer lattice points in k*C(n) where C(n) is a certain polytope with vertices having rational entries (the intersection of [0,1]^n with a hyperplane). Thus row n is an Ehrhart quasi-polynomial of degree n-1. - _Robert Israel_, Dec 12 2019
%H R. H. Hardin, <a href="/A200154/b200154.txt">Table of n, a(n) for n = 1..321</a>
%e Some solutions for n=7, k=6:
%e 5 6 5 3 6 0 0 5 4 1 2 2 0 2 1 2
%e 3 1 5 1 6 5 4 0 2 5 2 0 2 0 4 0
%e 3 3 6 5 6 1 6 2 0 1 1 4 3 4 6 2
%e 3 2 3 6 5 1 3 6 0 2 1 6 3 3 6 3
%e 2 0 2 5 5 3 2 6 1 6 2 5 3 1 5 2
%e 1 1 6 5 6 2 6 1 2 6 3 3 4 3 4 1
%e 4 1 1 3 1 2 0 1 5 0 3 1 6 1 2 4
%o (PARI) pad(d, n) = while(#d != n, d = concat([0], d)); d;
%o mydigits(i,n) = if (n<2, vector(i), digits(i,n));
%o bedt(n) = {for(i=2, #n=n, n=vecextract(n, "^1")-vecextract(n, "^-1")); n[1];}
%o T(n, k) = {k++; my(nbok = 0); for (i=0, k^n-1, d = pad(mydigits(i,k), n); if (bedt(d) == 0, nbok++);); nbok;} \\ _Michel Marcus_, Apr 08 2017
%Y Row 3 is A000982(n+1).
%Y Cf. A187202 (for 3rd PARI function).
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Nov 13 2011
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