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A183953
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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.
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17
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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
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OFFSET
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1,9
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COMMENTS
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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
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LINKS
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EXAMPLE
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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
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MAPLE
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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:
A183953rec(n, k, k*n^2) ;
end proc:
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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