A188002 T(n,k)=Number of nondecreasing arrangements of n numbers x(i) in -(n+k-2)..(n+k-2) with the sum of sign(x(i))*x(i)^2 zero

1, 1, 2, 1, 3, 3, 1, 4, 4, 12, 1, 5, 5, 17, 25, 1, 6, 8, 25, 45, 106, 1, 7, 9, 34, 64, 176, 376, 1, 8, 10, 46, 88, 286, 644, 1670, 1, 9, 11, 57, 117, 421, 1055, 2983, 7315, 1, 10, 12, 75, 159, 636, 1696, 5191, 13675, 35808, 1, 11, 15, 88, 216, 862, 2596, 8373, 24135, 67487, 176971

Offset

1,3

Links

R. J. Mathar, Table of n, a(n) for n = 1..434 augmenting an earlier file of 188 elements by R. H. Hardin

Example

Table starts
.....1.....1......1......1......1......1......1.......1.......1.......1......1
.....2.....3......4......5......6......7......8.......9......10......11.....12
.....3.....4......5......8......9.....10.....11......12......15......16.....17
....12....17.....25.....34.....46.....57.....75......88.....108.....125....147
....25....45.....64.....88....117....159....216.....270.....333.....421....500
...106...176....286....421....636....862...1206....1587....2114....2698...3450
...376...644...1055...1696...2596...3796...5443....7674...10392...14198..18641
..1670..2983...5191...8373..13343..20224..30358...43750...62354...86173.118859
..7315.13675..24135..40681..66452.105208.160866..242128..354103..510107.717077
.35808.67487.122238.211234.354806.573982.907542.1393159.2104002.3099873
Some solutions for n=5 k=3
.-4...-6...-5....0...-4...-1...-5...-3...-5...-5...-6...-6...-6...-4...-1...-4
.-2...-5...-5....0...-1....0...-2...-2....1....0....0...-6....3....2...-1...-1
.-2....3....3....0....2....0....2...-2....2....0....0....0....3....2...-1...-1
.-1....4....4....0....2....0....3....1....2....0....0....6....3....2...-1....3
..5....6....5....0....3....1....4....4....4....5....6....6....3....2....2....3

Maple

A188002rec := proc(n, nminusfE, E)
    option remember ;
    local a, fEminus, fEplus, f0 ;
    if E = 0 then
        if n = 0 then
            1;
        else
            0;
        end if;
    else
        a :=0 ;
        for fEminus from 0 to nminusfE do
            for fEplus from 0 to nminusfE-fEminus do
                f0 := nminusfE-fEminus-fEplus ;
                a := a+procname(n-E^2*fEminus+E^2*fEplus, f0, E-1) ;
            end do:
        end do:
        a ;
    end if;
end proc:
A188002 := proc(n, k)
    A188002rec(0, n, n+k-2) ;
end proc:
seq(seq( A188002(n, d-n), n=1..d-1), d=2..10) ; # R. J. Mathar, May 09 2023

Mathematica

f[n_, nminusfE_, E_] := f[n, nminusfE, E] = Module[{a, fEminus , fEplus, f0},     If[E == 0, If[n == 0, 1, 0], a = 0; For[fEminus = 0, fEminus <= nminusfE, fEminus++, For[fEplus = 0, fEplus <= nminusfE - fEminus, fEplus++, f0 = nminusfE - fEminus - fEplus; a = a + f[n - E^2*fEminus + E^2*fEplus, f0, E - 1]]]; a]];
T[n_, k_] := T[n, k] = f[0, n, n + k - 2];
Table[Table[ T[n, d - n], {n, 1, d - 1}], {d, 2, 12}] // Flatten (* Jean-François Alcover, Aug 21 2023, after R. J. Mathar *)

Prog

(PARI) A188002(n, k) = my(s, X, Y, p, pi, pj); s = (n+k-2)^2*n\2; Y = 'y + O('y^(s+1)); X = 'x + O('x^(n+1)); p = prod(i=1, n+k-2, 1/(1-X*Y^(i^2))); sum(i=0, n, pi=polcoef(p, i); sum(j=i, n-i, pj=polcoef(p, j); sum(d=0, s, polcoef(pi, d)*polcoef(pj, d)) * (2-(i==j)) )); \\ Max Alekseyev, Sep 18 2023

Crossrefs

Cf. A188003 (n=3), A188004 (n=4), A188005 (n=5), A188006 (n=6), A188007 (n=7), A188008 (n=8), A187994 (k=1), A187993 (k=n), A187995 (k=2), A187996 (k=3), A187997 (k=4), A187998 (k=5), A187999 (k=6), A188000 (k=7), A188001 (k=8).

Sequence in context: A211701 A183110 A117895 * A186974 A286312 A278492

Adjacent sequences: A187999 A188000 A188001 * A188003 A188004 A188005

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 18 2011

Status

approved