A200154 T(n,k) = number of 0..k arrays x(0..n-1) of n elements with zero (n-1)-st difference.

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, 25, 66, 103, 112, 45, 16, 1, 9, 32, 107, 202, 275, 182, 129, 6, 1, 10, 41, 158, 381, 730, 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

Offset: 1

Author

R. H. Hardin, Nov 13 2011

Keywords

• nonn
• tabl

Comments

Table starts

1 1 1 1 1 1 1 1 1 1 1

2 3 4 5 6 7 8 9 10 11 12

2 5 8 13 18 25 32 41 50 61 72

4 9 22 41 66 107 158 219 304 403 516

2 15 40 103 202 381 636 1033 1550 2287 3212

8 39 112 275 730 1589 3000 5181 8350 13871 21588

2 45 182 685 2036 5153 11370 23035 43284 76523 129052

16 129 688 2525 7488 18809 52166 121921 253768 484977 867086

6 149 844 5221 19262 68813 194818 514113 1171190 2531421 5019770

32 243 2090 13897 62772 256859 841122 2347671 6169890 14503751 31169760

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

Examples

			Some solutions for n=7, k=6:
  5  6  5  3  6  0  0  5  4  1  2  2  0  2  1  2
  3  1  5  1  6  5  4  0  2  5  2  0  2  0  4  0
  3  3  6  5  6  1  6  2  0  1  1  4  3  4  6  2
  3  2  3  6  5  1  3  6  0  2  1  6  3  3  6  3
  2  0  2  5  5  3  2  6  1  6  2  5  3  1  5  2
  1  1  6  5  6  2  6  1  2  6  3  3  4  3  4  1
  4  1  1  3  1  2  0  1  5  0  3  1  6  1  2  4
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..321

Crossrefs

Row 3 is A000982(n+1).

Cf. A187202 (for 3rd PARI function).

Programs

PARI

pad(d, n) = while(#d != n, d = concat([0], d)); d;
mydigits(i,n) = if (n<2, vector(i), digits(i,n));
bedt(n) = {for(i=2, #n=n, n=vecextract(n, "^1")-vecextract(n, "^-1")); n[1];}
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