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%I #25 Sep 19 2023 08:20:59
%S 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,
%T 1,8,10,46,88,286,644,1670,1,9,11,57,117,421,1055,2983,7315,1,10,12,
%U 75,159,636,1696,5191,13675,35808,1,11,15,88,216,862,2596,8373,24135,67487,176971
%N 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
%H R. J. Mathar, <a href="/A188002/b188002.txt">Table of n, a(n) for n = 1..434</a> augmenting an earlier file of 188 elements by _R. H. Hardin_
%e Table starts
%e .....1.....1......1......1......1......1......1.......1.......1.......1......1
%e .....2.....3......4......5......6......7......8.......9......10......11.....12
%e .....3.....4......5......8......9.....10.....11......12......15......16.....17
%e ....12....17.....25.....34.....46.....57.....75......88.....108.....125....147
%e ....25....45.....64.....88....117....159....216.....270.....333.....421....500
%e ...106...176....286....421....636....862...1206....1587....2114....2698...3450
%e ...376...644...1055...1696...2596...3796...5443....7674...10392...14198..18641
%e ..1670..2983...5191...8373..13343..20224..30358...43750...62354...86173.118859
%e ..7315.13675..24135..40681..66452.105208.160866..242128..354103..510107.717077
%e .35808.67487.122238.211234.354806.573982.907542.1393159.2104002.3099873
%e Some solutions for n=5 k=3
%e .-4...-6...-5....0...-4...-1...-5...-3...-5...-5...-6...-6...-6...-4...-1...-4
%e .-2...-5...-5....0...-1....0...-2...-2....1....0....0...-6....3....2...-1...-1
%e .-2....3....3....0....2....0....2...-2....2....0....0....0....3....2...-1...-1
%e .-1....4....4....0....2....0....3....1....2....0....0....6....3....2...-1....3
%e ..5....6....5....0....3....1....4....4....4....5....6....6....3....2....2....3
%p A188002rec := proc(n,nminusfE,E)
%p option remember ;
%p local a,fEminus, fEplus,f0 ;
%p if E = 0 then
%p if n = 0 then
%p 1;
%p else
%p 0;
%p end if;
%p else
%p a :=0 ;
%p for fEminus from 0 to nminusfE do
%p for fEplus from 0 to nminusfE-fEminus do
%p f0 := nminusfE-fEminus-fEplus ;
%p a := a+procname(n-E^2*fEminus+E^2*fEplus,f0,E-1) ;
%p end do:
%p end do:
%p a ;
%p end if;
%p end proc:
%p A188002 := proc(n,k)
%p A188002rec(0,n,n+k-2) ;
%p end proc:
%p seq(seq( A188002(n,d-n),n=1..d-1),d=2..10) ; # _R. J. Mathar_, May 09 2023
%t 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 T[n_, k_] := T[n, k] = f[0, n, n + k - 2];
%t Table[Table[ T[n, d - n], {n, 1, d - 1}], {d, 2, 12}] // Flatten (* _Jean-François Alcover_, Aug 21 2023, after _R. J. Mathar_ *)
%o (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
%Y 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).
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Mar 18 2011
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