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 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 16
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified August 10 16:53 EDT 2024. Contains 375058 sequences. (Running on oeis4.)