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%I #17 Jul 22 2022 12:13:20
%S 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,
%T 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,
%U 48,161,398,656,649,339,77,4,1,3,14,61,259,770,1613,2195,1805,771,118,2,1,4,17
%N 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.
%C 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
%H R. H. Hardin, <a href="/A183953/b183953.txt">Table of n, a(n) for n = 1..806</a>
%e Table starts
%e .1..1...1....1.....1.....1......1......1.......1.......1.......1........1
%e .1..1...1....2.....2.....2......2......3.......3.......3.......3........4
%e .1..2...2....3.....4.....6......7......9......12......14......17.......19
%e .1..1...4....8....14....21.....32.....48......61......82.....108......139
%e .2..5..10...27....53....94....161....259.....399.....578.....811.....1120
%e .1..7..26...78...180...398....770...1387....2330....3738....5772.....8599
%e .2.10..61..219...656..1613...3539...7099...13225...23247...38938....62599
%e .1.20.147..649..2195..6301..15601..34847...71509..137520..249799...433038
%e .3.37.339.1805..7250.23611..65909.163588..369777..775045.1525468..2847243
%e .3.77.771.4987.23044.85595.268008.737538.1830390.4178324.8894137.17852441
%e Some solutions for n=5
%e ..4....1....3....0....4....4....0....3....1....3....0....0....0....2....1....0
%e ..3....2....1....0....3....3....0....1....2....1....4....4....0....4....2....4
%e ..3....0....2....1....2....4....4....3....1....4....2....1....0....1....3....4
%e ..2....1....0....1....1....3....4....1....2....2....1....0....0....3....4....3
%e ..1....3....3....3....2....0....0....2....2....1....2....3....4....1....0....0
%p A183953rec := proc(n,k,s)
%p option remember;
%p local c;
%p if s < 0 then
%p return 0 ;
%p elif n = 0 then
%p if s =0 then
%p return 1;
%p else
%p return 0 ;
%p end if;
%p else
%p add( procname(n-1,k,s-c*n^2),c=0..k) ;
%p end if;
%p end proc:
%p A183953 := proc(n,k)
%p A183953rec(n,k,k*n^2) ;
%p end proc:
%p seq(seq( A183953(n,d-n),n=1..d-1),d=2..12) ; # _R. J. Mathar_, Mar 08 2021
%t 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 T[n_, k_] := r[n, k, k*n^2];
%t Table[Table[T[n, d-n], {n, 1, d-1}], {d, 2, 14}] // Flatten (* _Jean-François Alcover_, Jul 22 2022, after _R. J. Mathar_ *)
%Y Column 1 is A030273. A183946 (column 2), A183947 (column 3), A183954 (row 3), A183955 (row 4).
%K nonn,tabl
%O 1,9
%A _R. H. Hardin_, Jan 08 2011
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