A187991 -id:A187991 - cp's OEIS frontend

A187988 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))*2^|x(i)| zero.

0, 0, 1, 0, 2, 3, 0, 3, 5, 9, 0, 4, 7, 15, 36, 0, 5, 9, 22, 57, 117, 0, 6, 11, 30, 82, 181, 411, 0, 7, 13, 39, 111, 260, 632, 1452, 0, 8, 15, 49, 144, 355, 912, 2199, 5040, 0, 9, 17, 60, 181, 467, 1257, 3158, 7593, 17829, 0, 10, 19, 72, 222, 597, 1673, 4357, 10920, 26706, 62870

Offset: 1

R. H. Hardin, Mar 18 2011

Table starts
.....0.....0.....0.....0.....0.....0.....0.....0....0....0...0...0..0..0.0
.....1.....2.....3.....4.....5.....6.....7.....8....9...10..11..12.13.14
.....3.....5.....7.....9....11....13....15....17...19...21..23..25.27
.....9....15....22....30....39....49....60....72...85...99.114.130
....36....57....82...111...144...181...222...267..316..369.426
...117...181...260...355...467...597...746...915.1105.1317
...411...632...912..1257..1673..2166..2742..3407.4167
..1452..2199..3158..4357..5825..7592..9689.12148
..5040..7593.10920.15146.20404.26835.34588
.17829.26706.38385.53379.72246.95590

			Some solutions for n=5 k=3
.-3...-6...-5...-3...-3...-6...-4...-4...-1...-4...-2...-4...-4...-2...-3...-5
.-3...-3...-5...-1...-3...-5...-1...-2...-1...-4...-2...-2...-4...-1....1...-1
.-2...-3...-2....1...-3...-5....0...-2....0...-4...-1....2...-3....1....1....1
..2....4....2....2...-3....6....0....3....0...-4....1....3....3....1....1....4
..4....6....6....2....5....6....4....4....1....6....3....3....5....1....1....4

R. J. Mathar, Table of n, a(n) for n = 1..187 augmenting an earlier file with 117 entries by R. H. Hardin.
R. J. Mathar, Background on the recurrent Maple program for the linear diophantine equation

Row n=4 is A055999(k+1). A187989 (n=5), A187990 (n=6), A187991 (n=7), A187992 (n=8), A187979 (k=n), A187980 (k=1), A187981 (k=2), A187982 (k=3), A187983 (k=4), A187984 (k=5), A187985 (k=6).

AatE := proc(n,nminusfE,E)
    option remember ;
    local a,fEminus, fEplus,f0,resn ;
    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 ;
                resn := n-(2^E+1)*fEminus+(2^E-1)*fEplus ;
                if abs (resn) <= (1+2^(E-1))*f0 then
                    a := a+procname(resn,f0,E-1) ;
                end if;
            end do:
        end do:
        a ;
    end if;
end proc:
A187988 := proc(n,k)
    AatE(n,n,n+k-2) ;
end proc:
seq(seq( A187988(n, d-n), n=1..d-1), d=2..15) ; # R. J. Mathar, May 12 2023

AatE[n_, nminusfE_, E_] := AatE[n, nminusfE, E] = Module[{a, fEminus, fEplus, f0, resn}, 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; resn = n-(2^E+1)*fEminus + (2^E-1)*fEplus; If[Abs[resn] <= (1+2^(E-1))*f0, a = a + AatE[resn, f0, E-1]]]]; a]];
A187988[n_, k_] := AatE[n, n, n+k-2];
Table[Table[ A187988[n, d-n], {n, 1, d-1}], {d, 2, 15}] // Flatten (* Jean-François Alcover, Sep 18 2024, after R. J. Mathar *)

cp's OEIS Frontend

A187988 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))*2^|x(i)| zero.

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