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A184703
T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum i*x(i) equal to k*n.
17
1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 3, 6, 7, 3, 1, 3, 9, 14, 12, 4, 1, 4, 12, 28, 34, 21, 5, 1, 4, 17, 46, 78, 74, 35, 6, 1, 5, 22, 74, 156, 207, 154, 58, 8, 1, 5, 27, 107, 282, 476, 504, 304, 91, 10, 1, 6, 34, 154, 471, 985, 1349, 1169, 580, 142, 12, 1, 6, 41, 208, 744, 1842, 3142, 3571, 2574, 1066, 215, 15
OFFSET
1,5
COMMENTS
T(n,k) is the number of integer lattice points in k*P_n, where P_n is the polytope in R^n defined by the constraints 0 <= x_i <= 1 and Sum_{i=1..n} i x_i = n. Thus for each n, T(n,k) is an Ehrhart quasi-polynomial. - Robert Israel, Dec 21 2022
LINKS
EXAMPLE
Table starts:
1 1 1 1 1 1 1 1 1 1 1 1
1 2 2 3 3 4 4 5 5 6 6 7
2 3 6 9 12 17 22 27 34 41 48 57
2 7 14 28 46 74 107 154 208 278 357 456
3 12 34 78 156 282 471 744 1119 1623 2279 3118
4 21 74 207 476 985 1842 3226 5325 8414 12766 18789
5 35 154 504 1349 3142 6575 12688 22923 39266 64315 101460
6 58 304 1169 3571 9353 21713 46037 90595 167917 295811 499442
8 91 580 2574 8939 26146 67105 155645 332729 665317 1257898 2268061
10 142 1066 5439 21310 69331 195760 495251 1146377 2467215 4994696 9599863
Some solutions for n=5, k=4:
4 4 2 0 1 2 1 0 2 4 0 0 4 0 2 2
1 2 3 1 2 0 2 0 1 1 0 0 0 2 2 1
2 1 4 0 1 2 2 1 4 0 4 0 4 0 2 2
2 1 0 2 3 3 1 3 1 1 2 0 1 4 2 0
0 1 0 2 0 0 1 1 0 2 0 4 0 0 0 2
MAPLE
S:= proc(n, k, s) option remember; local j;
if n = 1 then
if s <= k then return 1 else return 0 fi
fi;
add(procname(n-1, k, s-j*n), j=0..min(s/n, k))
end proc:
[seq(seq(S(n, m-n, (m-n)*n), n=1..m-1), m=1..20)]; # Robert Israel, Dec 21 2022
MATHEMATICA
S[n_, k_, s_] := S[n, k, s] = Module[{}, If[n == 1, If[s <= k, Return@1, Return@0]]; Sum[S[n - 1, k, s - j*n], {j, 0, Min[s/n, k]}]];
Table[Table[S[n, m - n, (m - n)*n], {n, 1, m - 1}], {m, 1, 20}] // Flatten (* Jean-François Alcover, Aug 21 2023, after Robert Israel *)
CROSSREFS
Column 1 is A000009.
Row 3 is A008810(n+1).
Sequence in context: A236566 A350957 A046923 * A309287 A056173 A216817
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Jan 20 2011
STATUS
approved