A183953 T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum i^2*x(i) equal to k*n^2.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 3, 4, 5, 1, 1, 2, 4, 8, 10, 7, 2, 1, 2, 6, 14, 27, 26, 10, 1, 1, 3, 7, 21, 53, 78, 61, 20, 3, 1, 3, 9, 32, 94, 180, 219, 147, 37, 3, 1, 3, 12, 48, 161, 398, 656, 649, 339, 77, 4, 1, 3, 14, 61, 259, 770, 1613, 2195, 1805, 771, 118, 2, 1, 4, 17

Offset: 1

Author

R. H. Hardin, Jan 08 2011

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of integer lattice points in k*C(n) where C(n) is the polytope in R^n defined by the equation Sum_{1<=i<=n} i^2*x_i = n^2 and the inequalities 0 <= x_i <= 1. The vertices of the polytope have rational coordinates. Thus row n of the table is an Ehrhart quasi-polynomial of degree n-1. - Robert Israel, Jul 10 2019

Examples

			Table starts
.1..1...1....1.....1.....1......1......1.......1.......1.......1........1
.1..1...1....2.....2.....2......2......3.......3.......3.......3........4
.1..2...2....3.....4.....6......7......9......12......14......17.......19
.1..1...4....8....14....21.....32.....48......61......82.....108......139
.2..5..10...27....53....94....161....259.....399.....578.....811.....1120
.1..7..26...78...180...398....770...1387....2330....3738....5772.....8599
.2.10..61..219...656..1613...3539...7099...13225...23247...38938....62599
.1.20.147..649..2195..6301..15601..34847...71509..137520..249799...433038
.3.37.339.1805..7250.23611..65909.163588..369777..775045.1525468..2847243
.3.77.771.4987.23044.85595.268008.737538.1830390.4178324.8894137.17852441
Some solutions for n=5
..4....1....3....0....4....4....0....3....1....3....0....0....0....2....1....0
..3....2....1....0....3....3....0....1....2....1....4....4....0....4....2....4
..3....0....2....1....2....4....4....3....1....4....2....1....0....1....3....4
..2....1....0....1....1....3....4....1....2....2....1....0....0....3....4....3
..1....3....3....3....2....0....0....2....2....1....2....3....4....1....0....0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..806

Crossrefs

Column 1 is A030273. A183946 (column 2), A183947 (column 3), A183954 (row 3), A183955 (row 4).

Programs

Maple

A183953rec := proc(n,k,s)
    option remember;
    local c;
    if s < 0 then
        return 0 ;
    elif n = 0 then
        if s =0 then
            return 1;
        else
            return 0 ;
        end if;
    else
        add( procname(n-1,k,s-c*n^2),c=0..k) ;
    end if;
end proc:
A183953 := proc(n,k)
    A183953rec(n,k,k*n^2) ;
end proc:
seq(seq( A183953(n,d-n),n=1..d-1),d=2..12) ; # R. J. Mathar, Mar 08 2021

Mathematica

r[n_, k_, s_] := r[n, k, s] = Which[s < 0, 0, n == 0, If[s == 0, 1, 0], True, Sum[r[n-1, k, s-c*n^2], {c, 0, k}]];
T[n_, k_] := r[n, k, k*n^2];
Table[Table[T[n, d-n], {n, 1, d-1}], {d, 2, 14}] // Flatten (* Jean-François Alcover, Jul 22 2022, after R. J. Mathar *)