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.

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

R. H. Hardin, Table of n, a(n) for n = 1..2113

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

Adjacent sequences: A184700 A184701 A184702 * A184704 A184705 A184706

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 20 2011

Status

approved